**Name** | **Title** | **Date** |

Emmanuel Fleurentin (George Mason University and UNC Chapel Hill) | **Abstract:** Noise-induced tipping (N-tipping) emerges when random fluctuations prompt transitions from one (meta)stable state to another, potentially as a rare event. In this talk, we delineate new techniques for determining Most Probable Escapes Paths (MPEPs) in stochastic differential equations over periodic boundaries. We utilize a dynamical system approach to unravel MPEPs for the intermediate noise regime. We discuss the framework for computing the MPEPs by first looking at intersections of stable and unstable manifolds of invariant sets of a Hamiltonian system derived from the Euler-Lagrange equations of the Freidlin-Wentzell (FW) functional. The Maslov index helps identify which critical points of the FW functional are local minimizers and assists in explaining the effects of the interaction of noise and the deterministic flow. The Onsager-Machlup functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out the MPEP. We will illustrate our approach and compare our theoretical prediction with Monte Carlo simulations in the Inverted Van der Pol system and a carbon cycle model. Investigating Most Probable Escape Paths over Periodic Boundaries: a Dynamical Systems Approach | 18 June 2024 |

Juan Patino Echeverria (University of Auckland) | **Abstract:** Wild chaos is a form of higher-dimensional chaotic dynamics that can only arise in vector fields of dimension at least four. This talk explores wild chaos in a four-dimensional system of differential equations, which is an extension of the classic Lorenz equations. Recently, Gonchenko, Kazakov and Turaev (2021) showed, via the computation of Lyapunov exponents, that this system has a wild chaotic attractor at a particular point in parameter space.
To explain how this wild chaotic attractor arises geometrically, we perform a bifurcation analysis of the system in a two-parameter setting. As a starting point, we continue the one-parameter bifurcation structure of the classic Lorenz equations when the relevant new parameter is “switched on”. We find that the well-known homoclinic explosion point of the Lorenz system unfolds and gives rise to infinite cascades of curves of Shilnikov-type global connections in the four-dimensional system. These connections are formed by the unstable manifold of the origin, which plays an essential role in the emergence of complicated dynamics in the system. We also compute the kneading diagram that encodes how this one-dimensional manifold repeatedly moves around a pair of equilibria. In combination with the direct computation of curves of global bifurcations, the kneading diagram provides insight that helps identify regions where wild chaos may occur. Transitions to wild chaos in a four-dimensional Lorenz-like system | 4 June 2024 |

Tim Austin (University of Warwick) | **Abstract:** Entropy has its origins in thermodynamics and statistical mechanics. It gained mathematical rigour in Shannon's work on the foundations of information theory, and quickly found striking applications to ergodic theory in work of Kolmogorov and Sinai. Many variants and other applications have appeared in pure mathematics since, connecting probability, combinatorics, dynamics and other areas.
I will survey a few recent developments in this story, with an emphasis on some of the basic ideas that they have in common. I will focus largely on (i) Lewis Bowen's "sofic entropy", which helps us to study the dynamics of "large" groups such as free groups, and (ii) a cousin of sofic entropy in the world of unitary representations, which leads to new connections with random matrices.
This talk will be a fairly general survey. I will assume standard background in groups, measure theory and the language of probability, but only a basic awareness of ergodic theory. Notions of entropy in ergodic theory and representation theory | 14 May 2024 |

Carsten Wiuff (University of Copenhagen) | **Abstract:** This talk is about the dynamics and stationary behaviour of continuous time Markov chains on the non-negative integers. Such stochastic processes are abundant in applications, in particular in biology. There are two parts of the talk.
Part 1: Under certain conditions (mainly polynomial transition rates), we give threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, and positive/null recurrence.
Part 2: In the second part of the talk, we will characterize stationary distributions in terms of a few generating terms, which leads to a way to simulate the distribution efficiently. This is akin to known results for birth-death processes where there is one generating term
The Dynamics and Stationary Behaviour of 1d Markov Chains | 14 May 2024 |

Jonguk Yang (University of Zurich) | **Abstract:** A 1D smooth map on an interval is unimodal if it maps the interval into itself by folding it once (at the unique critical point). Analogously, a 2D smooth diffeomorphism on a square is Hénon-like if it maps the square into itself by squeezing it along the vertical direction to a thin strip, then bending it into a “C”-shape.
Joint with S. Crovisier, M. Lyubich and E. Pujals, we extended the celebrated renormalization theory of 1D unimodal maps to the 2D setting, so that it can be applied to the study of Hénon-like maps. In this talk, I will give an outline of our main results. This includes renormalization convergence, the uniqueness of the “2D critical point”, and the robustness of the required regularity conditions of the maps (so that they are finite-time checkable). Hénon-like Renormalization | 8 May 2024 |

Artur Avila (Universität Zürich) | **Abstract:** As discovered by Poincaré in the end of the 19th century, even small perturbations of very regular dynamical systems may display chaotic features, due to complicated interactions near a homoclinic point. In the 1960's Smale attempted to understand such dynamics in term of a stable model, the horseshoe, but this was too optimistic.Indeed, Newhouse showed that even in only two dimensions, a homoclinic bifurcation gives rise to particular wild dynamics, such as the generic presence of infinitely many attractors. This Newhouse phenomenon is associated to a renormalization mechanism, but also with particular geometric properties of some fractal sets within a Smale horseshoe. When considering two-dimensional complex dynamics those fractal sets become much more beautiful but unfortunately also more difficult to handle. Renormalization, Fractal Geometry and the Newhouse Phenomenon | 30 April 2024 |

Raphael Krikorian (Université de Cergy-Pontoise) | **Abstract:** Quadratic Hénon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h:(x,y)\mapsto (\lambda^{1/2}(x^2+c)-\lambda y,x)$. They have constant Jacobian equal to $\lambda$ and they admit two fixed points. If $\lambda$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that $h$ admits many quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, S. Ushiki observed some years ago what seems to be quasi-periodic orbits (though no Siegel disks exist). I will explain why this is the case. This theoretical framework also predicts (and proves), in the dissipative case ($\lambda$ of module less than 1), the existence of (attractive) Herman rings. These Herman rings, which were not observed before, can be produced in numerical experiments. Exotic rotation domains and Herman rings for quadratic Hénon maps | 30 April 2024 |

Irene De Blasi (University of Turin) | **Abstract:** A new type of billiard system, of interest for Celestial Mechanics, is taken into consideration: here, a closed refraction interface separates two regions in which different potentials (harmonic and Keplerian) act. The result is a variation of the classical Birkhoff billiard where the particle enters and exits from the domain, and can be used, for example, to mimic the motion of a particle in an elliptic galaxy having a central mass.
This model, which can be studied both in two and three dimensions, presents strong analogies with the more studied Kepler billiard, where a Keplerian inner potential is associated with a reflecting wall.
The dynamical properties of the system can be studied by adapting techniques coming from billiards’ theory, variational methods and results for general area-preserving maps, and regard principally the existence and stability properties of equilibrium trajectories or the arising of chaotic behaviours.
Work in collaboration with V. Barutello and S. Terracini.
References:
De Blasi I., Terracini S., Refraction periodic trajectories in central mass galaxies. Nonlinear Analysis (2022)
De Blasi I., Terracini, S., On some refraction billiards. Discrete and Continuous Dynamical Systems (2022)
Barutello V.L., De Blasi I., Terracini, S., Chaotic dynamics in refraction galactic billiards. Nonlinearity (2023)
Barutello V.L., De Blasi I., A note on chaotic billiards with potentials, Preprint (2023) Billiards with potentials in Celestial Mechanics: refractive case | 23 April 2024 |

Matthieu Astorg (University of Orléans) | **Abstract:** Bedford, Smillie and Ueda have introduced a notion of horn maps for polynomial diffeomorphisms of C2 with a semi parabolic fixed point, generalizing classical results from parabolic implosion in one complex variable. We prove that these horn maps satisfy a weak version of the Ahlfors island property. As a consequence, we obtain the density of repelling cycles in their Julia set, and we prove the existence of perturbations of the initial Hénon map for which the forward Julia set J^+ has Hausdorff dimension arbitrarily close to 4. Joint work with Fabrizio Bianchi. Horn maps of semi-parabolic Hénon maps | 9 April 2024 |

Nikolai Prokhorov (Aix-Marseille University) | **Abstract:** In the 1980’s, William Thurston obtained his celebrated characterization of post-critically finite rational maps. This result laid the foundation of such a field as Thurston's theory in holomorphic dynamics, which has been actively developing in the last few decades. One of the most important problems in this area is the characterization question, which asks whether a given topological map is equivalent to a holomorphic one. The result of W. Thurston and further developments allow us to answer this question quite effectively in the setting of (postcritically finite) maps of finite degree, and it has numerous applications for the dynamics of rational maps.
A similar question can be formulated for the maps of infinite degrees (i.e., in the transcendental setting), for instance, for entire postsingularly finite maps. However, the characterization problem becomes significantly more complicated, and the complete answer in the transcendental case is still not known.
In my talk, I am going to motivate the questions above and introduce the key notions of Thurston's theory in the transcendental setting. Further, I am going to explain the main techniques to attack the characterization problem and provide some examples. Finally, I am going to report about new classes of transcendental maps, where we managed to obtain an answer to this question. Towards Transcendental Thurston Theory | 9 April 2024 |

Felipe García-Ramos (UASLP & Jagiellonian University) | **Abstract:** This talk is an introduction to local entropy theory. We will review the basic results of the area and its applications and connections to other topics like homoclinic points, Li Yorke chaos, mean dimension and descriptive set theory. Local entropy theory | 26 March 2024 |

Davide Sclosa (Vrije Universiteit Amsterdam) | **Abstract:** Consider a set of dynamical units, placed on the vertices of a graph, and coupled by an odd function according to the graph edges. Systems of this form have emerged independently in opinion dynamics, control theory, and image processing. In such systems every trajectory converges to an equilibrium, but in general equilibria are not isolated. Indeed, we will show that in general the set of equilibria is union of manifolds of equilibria, which can intersect and have different dimension (even on asymmetric graphs). We will determine how the geometry of the set of equilibria is controlled by combinatorial aspects of the underlying graph (homology, coverings, connectivity,…) and discuss how to analyze stability along a manifold of equilibria. Graph Gradient Diffusion | 26 March 2024 |

Michele Coti Zelati (Imperial College London) | **Abstract:** We consider general two-dimensional autonomous velocity fields and prove that their mixing and dissipation features are limited to algebraic rates. As an application, we consider a standard cellular flow on a periodic box, and explore potential consequences for the long-time dynamics in the two-dimensional Euler equations. Diffusion and mixing for two-dimensional Hamiltonian flows
| 19 March 2024 |

Raphael Gerlach (Paderborn University) | **Abstract:** In this talk a framework for the global numerical analysis of infinite-dimensional dynamical systems is presented. By utilizing embedding techniques a dynamically equivalent finite-dimensional system, the so-called core dynamical system (CDS), is constructed. This system is then used for the numerical approximation of corresponding embedded invariant sets such as the embedded attractor or embedded unstable manifolds. Here, the focus lies on adapting set-oriented numerical tools, that generate coverings of the set of interest, to the CDS. As part of this, the subdivision scheme is also extended to parameter-dependent systems which allows to efficiently track a parameter-dependent attractor.
As the CDS heavily relies on the choice of an appropriate observation map, in the second part of this talk suitable numerical realizations of the CDS for delay differential equations and partial differential equations will be presented. Moreover for a subsequent geometric analysis a manifold learning technique called diffusion maps is considered, which reveals the intrinsic geometry of the embedded invariant set. In this context a set-oriented landmark selection scheme and an intrinsic dimension estimator is developed.
Finally, the numerical methods are applied to some well-known (infinite-dimensional) dynamical systems, such as the Mackey-Glass delay differential equation, the Kuramoto-Sivashinsky equation and plane Poiseuille flow. Glimpse of the Infinite: The Approximation of Invariant Sets of Infinite-Dimensional Systems | 12 March 2024 |

Laurent Stolovitch (Université de Nice Sophia Antipolis) | **Abstract:** In this work in collaboration with Z. Zhao (Nice), we consider analytic perturbations of isometries of an analytic Riemannian manifold $M$. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically conjugate on $M$ to the same group of isometry it is a perturbation of. Our result relies on a ``Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Our result generalizes Arnold-Herman's theorem about diffeomorphisms of the circle that are small perturbations of rotations. Local rigidity of actions of isometries on compact real analytic Riemannian manifolds | 12 March 2024 |

Isabelle Schneider (Freie Universität Berlin) | **Abstract:** Symmetries described by group transformations help immensely in our task to qualitatively characterize solutions of dynamical systems - providing that they exist. In this talk, we significantly enlarge the class of dynamical systems which can be studied by symmetry methods, moving our focus from groups to groupoids as the underlying algebraic structure describing symmetry. Building on the groupoid framework, we fundamentally generalize the notion of equivariance and equivariant bifurcation theory. We will also outline first applications in modelling and control of dynamical systems. Symmetry groupoids of dynamical systems | 5 March 2024 |

Christian Poetzsche (University of Klagenfurt) | **Abstract:** Integrodifference equations are successful models to describe spatial dispersal and temporary evolution. For this reason they can be understood as a discrete-time counterpart to reaction-diffusion equations and form an interesting class of infinite-dimensional dynamical systems. Nevertheless, for the sake of numerical simulations integrodifference equations require a spatial discretization.
In this talk, we investigate how their full hierarchy of invariant manifolds (stable, center- stable, center, center-unstable, unstable) behaves under the commonly used discretizations methods. We begin with the classical situation near periodic solutions and proceed to a general nonautonomous framework.
References:
[1] F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics, Springer, Cham, 2019.
[2] C. Pötzsche, Numerical dynamics of integrodifference equations: Periodic solutions and invariant manifolds in $C^\alpha(\Omega)$, Numerische Mathematik, 2023
[3] C. Pötzsche, Numerical dynamics of integrodifference equations: Hierarchies of invariant bundles in $L^p(\Omega)$, Numer. Funct. Anal. Optimization 44(7), 653-686, 2023 Numerical dynamics of integrodifference equations: From invariant manifolds to fiber bundles | 26 February 2024 |

Robert Skiba (Nicolaus Copernicus University) | **Abstract:** In this talk, I will provide a sufficient condition for bifurcation of
homoclinic solutions to nonautonomous ordinary differential equations.
This requires us to relate a crucial tool under the name of parity from
the abstract bifurcation theory of nonlinear Fredholm operators to
the Evans function, which was originally invented in the stability
theory of travelling waves for evolutionary differential equations.
This is a joint work with Christian Pötzsche.
References:
[1] P.M. Fitzpatrick, J. Pejsachowicz, Parity and generalized
multiplicity, Trans. Amer.Math.Soc.~326 (1991), 281--305.
[2] R. Skiba, C. Pötzsche, Evans function, parity and nonautonomous
bifurcation of bounded entire solutions (in preparations). Evans function, parity and nonautonomous bifurcation of homolcinic solutions | 26 February 2024 |

Dmitrii Mints (Imperial College London) | **Abstract:** Our research is aimed at studying the dynamics of smooth multidimensional diffeomorphisms from Newhouse domain, that is, open regions in the space of maps where systems with homoclinic tangencies are dense. We prove that maps with high order homoclinic tangencies of corank-1 (invariant manifolds forming the tangency have a unique common tangent vector) and maps having universal one-dimensional dynamics are dense in the Newhouse regions in the space of smooth and real-analytic multidimensional maps. We also show that in the space of smooth and real-analytic multidimensional maps in any neighborhood of a map such that it has bi-focus saddle periodic orbit whose invariant manifolds are tangent, there exist open regions in which maps with high order homoclinic tangencies of corank-2 (invariant manifolds forming the tangency have a plane of common tangent vectors) and maps having universal two-dimensional dynamics are dense.
High order homoclinic tangencies and universal dynamics for multidimensional diffeomorphisms | 20 February 2024 |

Hildeberto Jardon-Kojakhmetov (University of Groningen) | **Abstract:** In the context of dynamical systems, the blow-up method is a powerful technique that allows us to study nilpotent singularities of smooth differential equations. As an example, the blow-up method has been fundamental in the geometric analysis of slow-fast systems. In this talk, after a brief introduction to the method, I will discuss a couple of examples where the technique is also applied to slow-fast dynamical systems with a network structure and argue that the blow-up method may also be useful in that context. In the final part of the talk I digress about possible open-problems and applications. Blowing-up adaptive networks | 6 February 2024 |

Polina Vytnova (University of Surrey) | **Abstract:** A popular approach to Hausdorff dimension of limit sets relies on the Ruelle-Bowen formula, that connects the Hausdorff dimension to the spectral radius of a suitable nuclear operator. The spectral radius can then be approached via the dynamical zeta function or via the finite rank approximations. We will discuss functional analysis behind the latter in the setting of infinite hyperbolic iterated function systems arising from analytic families. Based on joint work with C. Wormell. Estimating the spectral radius of the transfer operator for an infinite IFS | 30 January 2024 |

David Lloyd (University of Surrey) | **Abstract:** In this talk, I present recent results on developing centre-manifold reduction methods for multidimensional localised patterns. In the first part of the talk, we concentrate on axisymmetric spikes emerging from the surface of a magnetic fluid placed between two Helmholtz coils that generate a uniform vertical magnetic field. It is shown how the ferrofluid problem (modelled by the Navier-Stokes equation coupled to Maxwell’s equations) can be decomposed into an infinite-system of coupled radial ODEs where invariant manifold theory can be applied to show the existence of various types of axisymmetric patterns. In the second half of the talk, I will then present an approximate theory for more complicated localised patches involving general dihedral cellular patterns again where radial invariant manifold theory can be used. Throughout the talk, several open analysis problems will be presented. This work is with Dan Hill, Matt Turner, and Jason Bramburger. Advances in the analysis of multidimensional localised patterns | 25 January 2024 |

Ale Jan Homburg (University of Amsterdam) | **Abstract:** This will be an informal talk on some novel and very simple constructions of systems with attractors that have intermingled basins. The idea is to use random walks along orbits of a given dynamical system Intermingled basins and thick attractors | 18 January 2024 |

Runze Zhang (Université de Toulouse) | **Abstract:** A rational function with complex coefficients is called hyperbolic if it is uniformly expanding over its Julia set. The dynamics on the Julia set remains topologically constant as the mapping varies within a hyperbolic component. However, the geometry of the Julia set undergoes a radical change as the mapping approaches the hyperbolic component boundary. It is therefore important to understand the structure of the boundary as well as the dynamics of the mappings on the boundary.
In this presentation, I focus on the family of cubic polynomials. Its main hyperbolic component is defined by the hyperbolic component containing $z^3$. Topologically, it is a real 4-ball, but its boundary exhibits a complicated fractal structure. I will discuss historical results and recent progress on this problem.
In collaboration with Jonguk Yang. Main hyperbolic component boundary for cubic polynomials | 12 December 2023 |

Nicolas Gourmelon (University of Bourdeaux) | **Abstract:** In this talk, I will introduce a notion of total renormalization - an example of which is the Rauzy induction - then I will build open sets of smooth diffeomorphisms that are totally renormalizable. I will explain how, concatenating such diffeormorphisms by surgery, we build a group $P$ of diffeormophisms that are total renormalizations of close to identity maps. Its tangent space at Identity is an (infinite dimensional) Lie algebra whose rigidity properties ultimately imply that $P$ is the group of diffeomorphisms isotopic to identity. This means that the smooth dynamics isotopic to identity are, in substance, the smooth dynamics close to identity. This answers questions by Takens-Ruelle, Turaev, Thouvenot. Joint work with Pierre Berger and Mathieu Helfter.
Diffeomorphisms isotopic to Identity "are dynamically" diffeomorphisms close to Identity. | 12 December 2023 |

Weikun He (Chinese Academy of Sciences) | **Abstract:** In this talk, we consider the action of a finitely generated group on the circle by analytic diffeomorphisms. We will discuss some results concerning the dimensions of objects arising from this action. More precisely, we will present connections among the dimension of minimal subsets, that of stationary measures, entropy of random walks, Lyapunov exponents and critical exponents. These can be viewed as generalizations of well-known results in the situation of PSL(2,R) acting on the circle. Dimension theory of groups of circle diffeomorphisms. | 7 December 2023 |

Leticia Pardo-Simón (University of Manchester) | **Abstract:** In this talk we will construct examples of transcendental entire functions with unbounded wandering domains, that is, Fatou components that are not eventually periodic. In particular, we provide examples of such functions with an orbit of unbounded fast escaping wandering domains, that is, that converge to infinity 'as fast as possible' under iteration of the map. Moreover, in relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering domains. For any given order greater than 1/2 and smaller than 1, we provide an entire function of such order with an unbounded wandering domain. This is based on joint work with A. Glücksam and V. Evdoridou. Unbounded fast escaping wandering domains | 28 November 2023 |

Maxime Breden (Ecole Polytechnique) | **Abstract:** The goal of a posteriori validation method is to get a quantitative and
rigorous description of some specific solutions of nonlinear dynamical systems, based on numerical simulations, which helps shed some light on the
global dynamics. The general strategy consists in combining a priori and
a posteriori error estimates, interval arithmetic, and a fixed point theorem
applied to a quasi-Newton operator. Starting from a numerically computed
approximate solution, one can then prove the existence of a true solution in
a small and explicit neighborhood of the numerical approximation.
In this talk I will present the main ideas behind these techniques, describe
a rather general framework in which they can be applied, and showcase
their interest by presenting examples of application related to population
dynamics, fluid dynamics and shear-induced chaos.
An introduction to computer-assisted proofs for dynamical systems: how to
turn a numerical simulation into a mathematical theorem. | 28 November 2023 |

Leon Staresinic (Imperial College London) | **Abstract:** Interval Translations Maps (ITM’s) are a natural generalisation of the well-known Interval Exchange Transformations (IET’s). They are obtained by dropping the bijectivity assumption for IET’s.
There are two basic types of ITM’s, finite-type and infinite-type ones. They are classified by their non-wandering sets: it is a finite union of intervals for finite-type maps and a Cantor set for infinite-type maps.
One of the basic questions in the field is: How prevalent is each type of map in the parameter space?
In this talk, we present work in progress in which we show that the set of stable finite-type maps forms an open and dense set in the parameter space of ITM’s with a fixed number of intervals.
This is joint work with Sebastian van Strien, Kostya Drach and Bjorn Winckler. Density of Stable Interval Translation Maps | 14 November 2023 |

Akshunna Shaurya Dogra (Imperial College London) | **Abstract:** Abstract: The approximation and generalization capacity of machine learning models has been profitably leveraged across a staggeringly wide variety of tasks. In particular, appropriately initialized Neural Networks sampled from suitable functional spaces invariably find stages of exponential learning. We introduce v - Tangent Kernels (vTKs), functional analytic objects partly inspired from the Neural Tangent Kernel (NTK), to build a generic theory for Neural Network optimization and generalization. Specifically, we prove that for sufficiently well-posed problems, appropriately initialized Neural Network models learn/solve tasks/problems at exponential/sub-exponential rates that are estimable before or during training. Notably, these results are showcased for a much wider class of loss functions/architectures than the standard mean squared error/large width regime that is usually the focus of conventional NTK analysis. The analysis applies to diverse practical problems solved using real networks such as differential equation solvers, shape recognition, classification, data fitting, feature extraction, etc. We exemplify the power of the vTK perspective by demonstrating the strong agreement between the predictions made from within this theory and the empirically observed optimization profiles, across different regimes and problems.
v Tangent Kernels | 7 November 2023 |

Alexey Korepanov (Loughborough University) | **Abstract:** I will talk about processes which appear in deterministic dynamical systems
such as Sinai billiards, where a point particle moves in a table with scatterers.
Often, on a suitable time scale, such processes can be approximated by a Brownian motion,
e.g. by constructing both on the same probability space so that they are almost surely close.
I'll talk about classical results as well as a very recent progress: a proof of approximation for
slowly mixing nonuniformly hyperbolic systems such as Bunimovich flowers, a joint work
with C.Cuny, J.Dedecker and F.Merlevede. Approximation by Brownian motion in deterministic dynamics | 31 October 2023 |

Giuseppe Tenaglia (Imperial College London) | **Abstract:** In this talk, we study a class of non random circle endomorphisms with additive noise, and show that, if the Lyapunov exponent is positive and a transitivity condition is satisfied, they present a random Young tower structure and quenched decay of correlations. Random Young towers and quenched decay of correlations for predominantly expanding multimodal circle maps | 17 October 2023 |

Caroline
Wormell (Australian National University) | **Abstract:** In applications it is often necessary to understand, predict or compress a dynamical system that is known only from data, such as a time series. This is commonly achieved using Dynamical Mode Decomposition (DMD) and its variants such as Extended DMD, which attempt to approximate the system's Koopman operator and spectrum by least squares approximation on the data. These algorithms are well-studied on stochastic and quasiperiodic dynamics, but little is known theoretically about their ouputs on chaotic systems.
Starting with a simple open class of chaotic maps, I show Extended DMD operator approximation relates closely to existing transfer operator theory. Proving a new result that least squares polynomial approximation with respect to a smooth, non-uniform measure is asymptotically as accurate as the usual Fourier approximation, I show that EDMD operator approximations can be up to exponentially accurate in the number of basis functions used. However, finite data effects severely destabilise the spectrum for deterministic chaos in a way that does not occur in stochastic systems. As a result, in practice, the number of basis functions used in EDMD may need to be limited, and only some of the spectrum of the operator obtained is realistically likely to be physically meaningful. Extended Dynamical Mode Decomposition and orthogonal polynomial approximation | 22 June 2023 |

Angxiu Ni (Beijing International Center for Mathematical Research) | **Abstract:** Physical measures encode the long-time statistics of chaotic dynamical systems, and the linear response is the parameter-derivative of physical measures. For physical measures of discrete-time hyperbolic chaotic systems, we give an equivariant divergence formula for the unstable perturbation of measure transfer operators along unstable manifolds. With this new formula, the linear response can be sampled by recursively computing only 2u many vectors on one orbit, where u is the unstable dimension. The numerical implementation of this formula is neither cursed by dimensionality nor the sensitive dependence on initial conditions. Our work generalizes conventional adjoint or backpropagation methods to chaotic systems. Adjoint method for chaos: sampling linear responses by an orbit | 22 June 2023 |

Jonguk Yang (University of Zurich) | **Abstract:** One of the most fundamental examples of non-linear dynamics is given by the class of unimodal interval maps. It is the simplest setting in which one can study the behavior of a critical orbit and the profound impact it has on the geometry of the system. By the works of Sullivan, McMullen and Lyubich, we have a complete renormalization theory for these maps, and as a result, their dynamics is now very well understood.
In this talk, we discuss the extension of this theory to a higher dimensional setting—namely, to properly dissipative diffeomorphisms in dimension two. Using the notion of non-uniform partial hyperbolicity, we identify what it means for such maps to be “unimodal.” Then we discuss how techniques from one-dimensional dynamics (namely, Denjoy lemma and Koebe distortion theorem) can be adapted to prove a priori bounds in this 2D setting.
This is based on a joint work with S. Crovisier, M . Lyubich and E. Pujals. A Priori Bounds for Unimodal Diffeomorphisms in Dimension Two | 6 June 2023 |

Gustavo Rodrigues-Ferreira (Imperial College London) | **Abstract:** Given a periodic orbit of a holomorphic map, the idea of relating its multiplier to the local dynamics of the map goes back to the nineteenth century. In the dynamical plane, it has led to many insights into the internal dynamics of periodic Fatou components. In the parameter plane, understanding how the multiplier changes can tell us many things about the structural stability and possible deformations of the map.
A wandering domain, however, has no periodic orbits, and though the possible internal dynamics can be classified through hyperbolic geometry, the corresponding parameter analysis has not been done. In this talk, we introduce distortion sequences of a wandering domain, an analogue of the multiplier of a periodic orbit, and show how it can (under certain hypotheses) be used to define a Banach-space-valued holomorphic map that functions as a multiplier map. Then, we will use quasiconformal surgery to discuss how to satisfy the aforementioned hypotheses. Distortion sequences of entire functions with wandering domains | 6 June 2023 |

Jordi Canela Sánchez (Universitat Jaume I) | **Abstract:** In this talk we will consider the dynamical system given by the iteration of a rational map Q over the Riemann Sphere. The dynamics of Q split the Riemann Sphere into two totally invariant sets. The Fatou set consists of all points z such that the family of iterates of Q is normal, or equivalently equicontinuous, in some open neighbourhood of z. The Fatou set is open and corresponds to the set of points with stable dynamics. Its complement, the Julia set, is closed and corresponds to the set of points which present chaotic behaviour.
Fatou components, connected components of the Fatou set, are mapped amongst themselves under iteration of Q. A periodic Fatou component can only have connectivity 1, 2, or infinity. Despite that, preperiodic Fatou components can have arbitrarily large finite connectivity. There exist explicit examples of rational maps with Fatou components of any prescribed connectivity. However, the degree of these maps grows as the required connectivity increases.
We study a family of singular perturbations of rational maps with a single free critical point. Under certain conditions, the dynamical planes of these singular perturbations contain Fatou components of arbitrarily large finite connectivity. In this talk we will analyze the dynamical conditions under which these Fatou components of arbitrarily large connectivity appear. Achievable connectivities of Fatou components in singular perturbations | 23 May 2023 |

Cristina Stoica (Wilfrid Laurier University) | **Abstract:** I will discuss a general framework for defining restricted problems in mechanical systems with symmetry. A typical example is that of the restricted three body problem in celestial mechanics. I will state a theorem concerning the persistence of dynamical features from restricted to non-restricted problems. The techniques used are based on scaling, symplectic reduction and the description of the linearisation operator at a relative equilibrium.
I will present some applications to systems such as a gravitationally coupled rigid body and a point mass, and the spherical double pendulum with a small mass at the free end. A continuation theorem in classical mechanics | 2 May 2023 |

Tanya Schmah (University of Ottawa) | **Abstract:** Image registration, i.e. alignment, is a fundamental problem in computer vision, including in medical imaging. While deep learning has made an enormous impact on this problem, the gold standard is still the geometric, or variational, approach which is based on geodesic motions in a diffeomorphism group (or in the group orbit of a particular image).
We consider geodesic motions in a hierarchy of Lie subgroups of a diffeomorphism group, with applications to image registration. By defining a Riemannian metric in terms of the associated Lie algebra decomposition, we can favour simpler transformations, for example affine or volume-preserving. By varying the metric over the course of an iterative method such as geodesic shooting, we can gradually add complexity, suggesting an alternative to multi-scale methods. Geodesic motions in Lie group hierarchies, with applications to image registration | 2 May 2023 |

Kostya Drach (IST Austria) | **Abstract:** In one-dimensional complex dynamics, a branch of dynamical systems that studies iterations of holomorphic maps, the rigidity question has a classical form: Under which conditions can one promote topological conjugacy between a pair of maps to an analytic (conformal) conjugacy? We call this 'parameter rigidity', as it allows us to distinguish maps within the parameter space starting with some 'soft' topological (or even combinatorial) data. On the other hand, for a given map, there is a parallel 'dynamical rigidity' question: Can one distinguish individual orbits in combinatorial terms (e.g. via symbolic dynamics)? For polynomials, and especially for quadratic polynomials without neutral periodic points, this circle of questions is well-studied in the works of Avila, Douady, Hubbard, Lyubich, McMullen, Sullivan, van Strien, Yoccoz, and many others. In this case, progress was possible thanks to the fact that most polynomials without neutral periodic points have a naturally defined Markov partition in their dynamical plane. It is a priori unclear how to construct such partitions for polynomials with neutral periodic points, or for general rational maps.
In the talk, I will discuss what we know so far about dynamical and parameter rigidity of general rational maps. Our guiding example will be Newton maps, a family of rational maps naturally arising from Newton's root-finding method. I will also outline a 'toolbox' of techniques designed to attack such rigidity questions; the key 'tool' is a generalized renormalization concept of complex box mapping. Time permitting, I will present some recent progress (jointly with Jonguk Yang) on rigidity of polynomials with bounded-type Siegel disks, where the same ‘toolbox’ is being used. Rigidity of rational maps | 18 April 2023 |

Adam Spiewak (Polish Academy of Sciences) | **Abstract:** We study prediction of dynamical systems from measurements performed via a one-dimensional observable along an orbit of the system. We give new versions of the Takens time-delay embedding theorem, both in the deterministic and probabilistic setting. In the latter case (when self-intersections in the reconstructed attractor can occur) we obtain upper bounds on the decay rate of prediction errors, as conjectured by Schroer, Sauer, Ott and Yorke. This is joint work with Krzysztof Baranski and Yonatan Gutman. Prediction of dynamical systems from time-delayed measurements with self-intersections | 28 March 2023 |

Rainer Klages (Queen Mary University of London) | **Abstract:** A random dynamical system consists of a setting where different types of dynamics are generated randomly in time. Here we consider as a simple example a piecewise linear one-dimensional map on the unit interval, where an irregular or a regular dynamic is randomly selected at each discrete time step for determining the outcome at the next time step. By continuously varying as a control parameter between zero and one the sampling probability to select the one or the other dynamic, the completely irregular, respectively regular dynamics are recovered as limiting cases. Hence, there must be a transition between these two states, where the combined dynamics exhibit intermittency. In my talk I will explore this transition in detail. I will show that there is a critical sampling probability at which the dynamic becomes anomalous, in the sense that the invariant density is non-normalisable, there is loss of ergodicity, and the correlations decay as a power law. I will also explore the impact of this transition onto the diffusive properties of a related lifted map.
[1] J.Yan, M.Majumdar, Y.Sato, S.Ruffo, C.Beck, R.Klages, preprint
[2] Y.Sato, R.Klages, Phys.Rev.Lett. 122, 174101 (2019) Transition to anomalous dynamics in simple random maps | 21 March 2023 |

Manjunath Gandhi (University of Pretoria) | **Abstract:** Reservoir computing systems are dynamical systems driven exogenously by an input sequence. In the machine learning world, they are used for information processing of data that arrives in streams. In particular, the dynamics of such systems can be exploited for generating observables for forecasting and modeling dynamical systems and entail a degree of stability that is missing in Takens delay embedding. The talk would center on the topological dynamics of such systems for applications, and also statistical properties for potential applications. Topological and Statistical Properties of Reservoir Computing Systems (online) | 14 March 2023 |

Aneta Stefanovska (Lancaster University) | **Abstract:** Physics distinguishes between isolated systems (no exchange of energy or matter with the environment), and non-isolated systems. The latter can either be closed, only able to exchange energy with the environment, or open in which case they can exchange both energy and matter with the environment. Already by 1943 during his lectures delivered under the auspices of the Dublin Institute for Advanced Studies where he was Director of Theoretical Physics, at Trinity College, Dublin, Erwin Schrödinger pointed out that
...living matter, while not eluding the "laws of physics" as established up to date, is likely to involve "other laws of physics" hitherto unknown, which however, once they have been revealed, will form just as integral a part of science as the former.
Schrödinger furthermore clarified [1] that living matter evades the decay to thermodynamic equilibrium by homeostatically maintaining negative entropy in an open system.
However, even today, we do not have the new laws of physics needed to describe living systems. Open systems are usually treated within the framework of statistical physics and stochastic dynamics, and the data measured from such systems are increasingly analysed using artificial intelligence.
In this talk we will propose oscillatory non-autonomous dynamical systems [2] as potential candidates to describe open systems. The talk will consist of four parts. First, we will consider several examples on a variety of timescales and size scales - including cellular systems up to the cardiovascular and brain dynamics, as well as diurnal oscillations and population levels [3-6], to illustrate the motivation for our proposal. Secondly, we will introduce chronotaxic (chronos - time, taxis - order) systems [7], proposed to provide a framework for non-autonomous systems that can endure continuous external perturbation, focusing on the properties of frequency-modulated dynamical systems and networks. We will illustrate that oscillators subject to driving with a slowly varying frequency, counterintuitively, have enlarged Arnold tongue in the parameter space [8]. Thirdly, we will present an experimental model of electrons on the surface of liquid helium subject to an electromagnetic field and microwaves that exhibit chronotaxic dynamics [9]. So far, chronotaxic dynamics has been observed mainly in living systems allowing only very limited possibilities for studying experimentally the effect of changing relevant parameter values. In contrast to these biological observational studies, the superfluid helium experiments offer us an opportunity to gain an understanding of chronotaxic behavior based on well-controlled laboratory investigations that promise to reveal fully the relationship between the physical world and the fundamental nature of chronotaxic dynamics. Lastly, we will introduce the idea of describing living systems as networks of networks of chronotaxic systems. We will propose that the conservation of phase difference could be the new law of physics that underlines living systems. Is non-autonomous dynamics in mathematics sufficient to describe open systems in physics? Lessons from cellular, cardiovascular and brain dynamics | 10 March 2023 |

Dan Wilson (University of Tennessee) | **Abstract:** I will discuss recent work that considers the perturbed behavior of a general dynamical system relative to a continuous family of either limit cycles or fixed points that emerge for different parameter sets. Using an appropriately defined isostable-based coordinate system which encodes for level sets of the slowest decaying eigenfunctions of the Koopman operator, this approach yields analytically tractable reduced order models that are valid in the strongly perturbed regime. In other words, the resulting dynamical models do not require restrictive order epsilon limitations on the magnitude of the perturbations. In practice, this approach allows for the consideration of large magnitude inputs in situations where other standard model identification and model order reduction techniques fail. Applications involving phase resetting of circadian rhythms following rapid travel across multiple time zones and phase locking of neural rhythms in response to strong synaptic coupling illustrate the utility of these new methods.
Associated data-driven methods for inference of phase-isostable-based models will also be discussed for use when the underlying dynamical equations are unknown or unavailable. In these examples, the proposed strategies outperform a collection of other commonly used data-driven model identification algorithms including Koopman model predictive control, Galerkin projection, and DMDc. Model Order Reduction Using Adaptive Phase and Isostable Coordinates | 7 March 2023 |

Iacopo Longo (TU Munich) | **Abstract:** This talk deals with rate- and size-induced tipping in nonautonomous scalar concave coercive differential equations. We show that, for this class of problems, the only possible bifurcation is the nonautonomous saddle-node bifurcation and that all tipping points occurring for these equations are in fact bifurcations of such type.
Moreover, the detailed description of the critical transition is completed by rigorous and calculable criteria to identify the tipping and tracking scenarios without relying on the numerical approximation of locally pullback attractive solutions.
This is a joint work with Carmen Núñez and Rafael Obaya from the University of Valladolid, Spain. A bifurcation approach to rate-induced tipping for concave coercive scalar ODEs | 7 March 2023 |

Jinxin Xue (Tsinghua University) | **Abstract:** It is classically known in Nielson-Thurston theory that the mapping class group of a hyperbolic surface is generated by Dehn twists and most elements are pseudo Anosov. Pseudo Anosov elements are interesting dynamical objects. They are featured by positive topological entropy and two invariant singular foliations expanded or contracted by the dynamics. We explore a generalization of these ideas to symplectic mapping class groups. With the symplectic Dehn twists along Lagrangian spheres introduced by Arnold and Seidel, we show in various settings that the compositions of such twists have features of pseudo Anosov elements, such as positive topological entropy, invariant stable and unstable laminitions, exponential growth of Floer homology group, etc. This is a joint work with Wenmin Gong and Zhijing Wang. Dynamics of composite symplectic Dehn twists (online) | 28 February 2023 |

Ale Jan Homburg (University of Amsterdam and Free University Amsterdam) | **Abstract:** We analyze the two-point motions of iterated function systems on the
unit interval generated by expanding and contracting affine maps, where
the expansion and contraction rates are determined by a pair $(M,N)$ of
integers.
This dynamics depends on the Lyapunov exponent.
For a negative Lyapunov exponent we establish synchronization, meaning
convergence of orbits with different initial points. For a vanishing
Lyapunov exponent we establish intermittency, where orbits are close for
a set of iterates of full density, but are intermittently apart. For a
positive Lyapunov exponent we show the existence of an absolutely
continuous stationary measure for the two-point dynamics and discuss its
consequences.
For nonnegative Lyapunov exponent and pairs $(M,N)$ that are
multiplicatively dependent integers, we provide explicit expressions for
absolutely continuous stationary measures of the two-point dynamics.
These stationary measures are infinite $\sigma$-finite measures in the
case of zero Lyapunov exponent.
This is joint work with Charlene Kalle. Iterated function systems of linear expanding and contracting maps on the unit interval | 28 February 2023 |

Adam Dor-On (Haifa University) | **Abstract:** The ratio-limit boundary \partial_{\rho} G for a random walk on G is defined as the remainder obtained after compactifying G with respect to ratio-limit kernels H(x,y) = \lim_n \frac{P^n(x,y)}{P^n(e,y)}, where P^n(x,y) is the n-th transition probability to pass from x to y. These limits were shown to exist for various classes of groups, normally by establishing a local limit theorem which measures the asymptotic behavior of P^n(x,y) as n \rightarrow \infty. In increasing level of generality, deep works of Woess, Lalley, Gouzel and Dussaule establish such local limit theorems for symmetric random walks on relatively hyperbolic groups. Techniques developed in these works then allow us to study \partial_{\rho} G. For instance, when G is hyperbolic, Woess applied techniques of Gouzel to show that \partial_{\rho} G is the Gromov boundary of G.
In this talk we will explain how to show that for a large class of random walks on relatively hyperbolic groups, the ratio-limit boundary is essentially minimal. That is, there is a unique minimal closed G-invariant subspace of \partial_{\rho} G. This result is motivated by applications in operator algebras, and indeed, by using it we are able to show the existence of a co-universal quotient of Toeplitz C*-algebras for such random walks. These co-universal C*-algebras are ubiquitous in the literature, going back to works of Cuntz and Krieger, and identifying them in various scenarios is fundamental for understanding the structure theory of C*-algebras. In the talk we will focus mostly on dynamics, topology and geometry, and if time permits I will explain the operator algebraic motivation and results.
*This talk is based on joint work with Matthieu Dussaule and Ilya Gekhtman. Essential minimality of ratio-limit boundary for random walks | 21 February 2023 |

Zhiyuan Zhang (Sorbonne Paris Nord University) | **Abstract:** In a work in progress with Avila and Lyubich, we show that
there are maps in the complex Hénon family with a stable homoclinic
tangency. This is due to a new mechanism on the stable intersections
between two dynamical Cantor sets generated by two classes of conformal
IFSs on the complex plane. Newhouse phenomenon in the complex Hénon family (online) | 14 February 2023 |

Genady Levin (The Hebrew University of Jerusalem) | **Abstract:** The Ruelle-Thurston transfer operator for rational maps. An explicit example: disconnected quadratic Julia sets
and the limit distribution of eigenvalues. Milnor-Thurston's and Tsujii's approaches to monotonicity of entropy in the real quadratic family.
A local approach: holomorphic deformation of a marked map, lifting of holomorphic motions, local transfer operator and its spectrum.
The main theorem and some applications. A critically infinite case: do saddle-nodes unfold in a positive direction?
Based partially on early works with Alex Eremenko, Mikhail Sodin and Peter Yuditskii, and recent works with Weixiao Shen and Sebastian van Strien. Transfer operator and its application to monotonicity for interval maps | 7 February 2023 |

Ivan Ovsyannikov (University of Hamburg) | **Abstract:** Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robustly chaotic, i.e. they preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist in applied models and be observed in experiments. It is known that discrete Lorenz attractors can appear in local and global bifurcations of multidimensional diffeomorphisms. The first such result was established for a certain three-dimensional Henon map, -- a quadratic map with a constant Jacobian.
In this presentation, I discuss the possibility for the inverse map of this Henon map also to have discrete Lorenz attractors. The idea is to find a fixed or periodic point with eigenvalues (-1, -1, +1) and check the normal form coefficients. The main result is a numerically established existence of a period-six orbit with the required linear part. Then the sixth iteration of the map can be approximated by a flow that is equivalent to the Shimizu-Morioka system. This means that the map and its inverse both can possess discrete Lorenz attractors. Wild Lorenz Attractors in a Three-Dimensional Hénon Map and in Its Inverse | 31 January 2023 |

Sajjad Bakrani (NODDS Lab, Kadir Has University) | **Abstract:** We consider a \(\mathbb{Z}_2\)-equivariant flow in \(\mathbb{R}^{4}\) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit \(\Gamma\). We provide criteria for the existence of stable and unstable invariant manifolds of \(\Gamma\). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic, there exist infinitely many multi-pulse homoclinic loops. This is a joint work with Dmitry Turaev and Jeroen Lamb.
Invariant manifolds of homoclinic orbits: super-homoclinics and multi-pulse homoclinic loops | 31 January 2023 |

Eddie Nijholt (Imperial College London) | **Abstract:** We define so-called network multipliers: typically small, matrix-valued expressions in the coefficients of linear network maps, that together describe the spectrum of general linear maps for the network in question, as well as for a whole range of derived networks. Due to their properties, these network multipliers can often be found by considering "smart choices" of networks. This is joint work with Prof. Lee DeVille from UIUC. Linear formulas for eigenvalues of network Jacobians | 24 January 2023 |

Dongchen Li (Imperial College London) | **Abstract:** We show that any system having a heterodimensional cycle can be approximated in C^r (r>1) topology by robust heterodimensional cycles, which implies a counterpart to the Newhouse theorem for homoclinic tangencies. The result is based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders. Persistence of heterodimensional cycles | 24 January 2023 |

Alexandre Rodrigues (University of Porto) | **Abstract:** In this informal talk, I discuss the stability of cycles within a heteroclinic network formed by different cycles, for a one-parameter model developed in the context of game theory. I describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map -- the projective map. Stability of the fixed points of the projective map determines the stability of the associated cycles. All concepts will be gently introduced and the talk will be accessible to non-specialists. This is a joint work with Telmo Peixe (ISEG, CEMAPRE). Stability of heteroclinic cycles: a new approach | 13 December 2022 |

Timothée Benard (University of Cambridge) | **Abstract:** We prove the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of a walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures. Joint work with Emmanuel Breuillard. Local limit theorem on nilpotent Lie groups | 6 December 2022 |

Yi Shi (Peking University) | **Abstract:** In this talk, we address the strong rigidity properties from joint integrability in the setting of Anosov diffeomorphisms on tori. More specifically, for an irreducible Anosov diffeomorphism with split stable bundle, the joint integrability of the strong stable and full unstable subbundles implies the existence of fine dominated splitting along the weak stable subbundle as well as Lyapunov exponents rigidity. This builds an equivalence bridge between the geometric rigidity (joint integrability) and dynamical spectral rigidity (Lyapunov exponents rigidity) for Anosov diffeomorphisms on tori.
This talk is based on a joint work with A. Gogolev.
Spectral rigidity and joint integrability for Anosov diffeomorphisms on tori (Zoom talk)
| 29 November 2022 |

Elena Queirolo (Technische Universität München) | **Abstract:** Coming from cell biology, a switching system is an ODE of the form $\dot x = - \gamma x + H(p, x)$, where $x$ is a vector and $H$ is a step function w.r.t. to each coordinate of $x$. Usually, the parameter space is high dimensional, adding a level of complexity to our study.
We first briefly present a method to fully determine the dynamical behaviour of some switching systems both in phase space and parameter space. Then, we discuss what happens when $H$ is smoothened: for which parameters $p$ does the dynamical behaviour change? When is it stable w.r.t. smoothening? Numerical approaches are developed to carry on extensive statistical studies. Smoothening switching systems (Zoom talk)
| 22 November 2022 |

Lukas Gonon (Imperial College London) | **Abstract:** In this talk we provide an introduction to reservoir computing and present our recent results on its mathematical foundations. Motivated by their performance in applications -- ranging from realized volatility forecasting to chaotic dynamical systems -- we study approximation and learning based on random recurrent neural networks and more general reservoir computing systems. Using techniques from statistical learning theory we obtain high-probability bounds on the approximation error and generalization error bounds for weakly dependent (possibly non-i.i.d.) input data.
The talk is based on joint works with Christa Cuchiero, Lyudmila Grigoryeva, Juan-Pablo Ortega and Josef Teichmann. Mathematical foundations of dynamic learning based on reservoir computing
| 22 November 2022 |

Matheus de Castro (Imperial Colledge London) | **Abstract:** Given an absorbing Markov chain $\{X_n\}_{n\in \mathbb N}$ on $M\sqcup \partial,$ absorbed at $\partial$ (i.e. $X_n =\partial $ implies $X_{n+1} = \partial)$. The statistical long-term behaviour of paths that remains in $M$ over a long time can be described by the limits
$$\lim_{n\to\infty} \mathbb P \left[X_n\in A \mid X_0 = x ,\tau >n\right] := \lim_{n\to\infty} \frac{\mathbb P \left[X_n\in A\mid X_0 = x\right]}{\mathbb P \left[X_n\in M\mid X_0 = x\right]}$$
and
$$\lim_{n\to\infty}\mathbb E \left[\frac{1}{n} \sum_{i=0}^{n-1} \mathbf{1}_A \circ X_i \hspace{0.1cm}\bigg\vert\hspace{0.1cm} X_0=x,\tau >n\right],$$
for a given measurable set $A \subset M.$
Using Banach lattice techniques, we establish an intrinsic relation between the above two limits. Moreover, we apply these results to the absorbing Markov chain $Y_{n+1} = \omega_n Y_n (1-Y_n),$ on $\mathbb R$ absorbed in $\partial = \mathbb R\setminus [-1,1]$, where $\{\omega_n\}_{n\in\mathbb N}$ is an i.i.d. sequence of random variables such that $\omega_n\sim \mathrm{Unif}([a,b]),$ with $1\leq a <4\leq b$.
This talk is based on a joint work with Martin Rasmussen, Jeroen S. W. Lamb and Vincent P.H. Goverse. Quasi-ergodic measures for absorbing Markov chains with applications to random logistic maps with escape | 15 November 2022 |

Matteo Tanzi (Laboratoire de Probabilités, Statistique et Modélisation) | **Abstract:** Recently, much progress has been made in the mathematical study of self-consistent transfer operators describing the thermodynamic limit of globally coupled expanding maps. Existence of equilibrium measures (fixed points for the self-consistent transfer operator) has been established together with their stability under perturbations and linear response. One of the main questions remaining open is to which extent the thermodynamic limit describes the evolution of the finite dimensional system.
More precisely, I will consider N uniformly expanding coupled maps when N is finite but large. I will introduce self-consistent transfer operators that approximate the evolution of measures under the dynamics, and quantify this approximation explicitly with respect to N. Using this result, I will prove that uniformly expanding coupled maps satisfy propagation of chaos when N tends to infinity, and characterize the absolutely continuous invariant measures for the finite dimensional system. The main working assumption is that the expansion is not too small and the strength of the interactions is not too large, although both can be of order one. In contrast with previous approaches, we do not require the coupled maps and the interactions to be identical. The technical advances that allow us to describe the system are: the introduction of a framework to study the evolution of conditional measures along some non-invariant foliations where the dependence of all estimates on the dimension is explicit; and the characterization of an invariant class of measures close to products that satisfy exponential concentration inequalities. Uniformly Expanding Coupled Maps: Self-Consistent Transfer Operators and Propagation of Chaos
| 8 November 2022 |

Lorenzo Diaz (Pontifical Catholic University of Rio de Janeiro) | **Abstract:** This is a somewhat philosophical talk (without theorems) about the notion of an elementary piece of dynamics and its generation. The emphasis will put on constructions. The works of Newhouse in the 70's stated that the locally generic coexistence of infinitely many sinks at (dissipative) homoclinic tangencies. We first study how this question can be adapted to heterodimensional cycles in partially hyperbolic settings (where sinks/sources cannot be displayed). While sinks are clearly independent pieces of dynamics, the first step is to discuss "what is (or what should be) an independent piece of dynamics" (with an special emphasis in the partially hyperbolic context). We propose that homoclinic classes play such a role. Thereafter we will explain Newhouse's construction of diffeomorphisms with infinitely many sinks. This follows the inductive pattern: a tangency leads to a sink and a new tangency. We will see that this sort of inductive construction cannot lead to "infinitely many independent homoclinic classes" and explain the obstructions. Against (or about) the generation of (infinitely many) independent homoclinic classes at heterodimensional cycles | 2 November 2022 |

Pablo Barrientos (Universidade Federal Fluminense) | **Abstract:** For random compositions of independent and identically distributed measurable maps on a Polish space, we study the existence and finitude of absolutely continuous ergodic stationary probability measures (which are, in particular, physical measures) whose basins of attraction cover the whole space almost everywhere.
We characterize and hierarchize such random maps in terms of their associated Markov operators, as well as show the difference between classes in the hierarchy by plenty of examples, including additive noise, multiplicative noise, and iterated function systems. We also provide sufficient practical conditions for a random map to belong to these classes. For instance, we establish that any continuous random map on a compact Riemannian manifold with absolutely continuous transition probability has finitely many physical measures whose basins of attraction cover Lebesgue almost all the manifold.
Finitude of physical measures for random maps | 2 November 2022 |

Dongchen Li (Imperial College London) | **Abstract:** A blender is a hyperbolic basic set such that non-transverse intersections with its invariant manifolds can be unremovable by small perturbations. In this talk, we consider blenders in the symplectic setting and show that, under certain generic conditions, symplectic blenders exist arbitrarily close to any whiskered torus that has a homoclinic connection. Symplectic blenders | 2 November 2022 |

Dmitry Turaev (Imperial College London) | **Abstract:**
Shilnikov scenario of the transition to chaos reduces to the study of the dynamics of local diffeomorphisms of a punctured disc. We show that such maps can have pseudohyperbolic (i.e., robustly chaotic) attractors (stable chain-transitive sets) which exhibit robustly heterodimensional dynamics, i.e., the orbits with different numbers of positive Lyapunov exponents coexist within the chain-transitive attractor and this property persists at small perturbations.
A model of a pseudohyperbolic Shilnikov attractor
| 2 November 2022 |

Julian Newman (University of Exeter) | **Abstract:** We make use of a certain one-dimensional stabilisation phenomenon previously described in the physics literature to provide a simple and yet stark illustrative example of how classical mathematical definitions of dynamical stability properties (based on limiting behaviour as time tends to infinity) can be deficient for describing qualitative dynamical stability; and motivated by this example, we develop a new framework for defining and describing stability properties of finite-time systems subject to slowly time-dependent forcing. Namely, our approach is to formulate the dynamics as a slow-fast system in which the slow time is constrained to a compact interval, and define stability properties somewhat analogously to the classical definitions except with the "infinite limit" being in the timescale separation rather than in the infinite progression of time. We obtain rigorous results for the case of one-dimensional systems (which includes our original example). Finite-time stabilisation by slow-timescale forcing | 18 October 2022 |

Eddie Nijholt (Imperial College London) | **Abstract:** Network dynamical systems appear abundantly in both nature and
engineering. A major obstacle in their analysis is the fact that most
established techniques from dynamical systems theory are not suited to
deal with them, as they involve coordinate transformations that destroy
the network structure. We present a novel solution, by showing that many
network properties -and sometimes the network structure itself- can be
realized as a form of symmetry. The key is to move away from the
classical notion of group symmetry, and to consider more exotic
algebraic structures such as quivers instead. This results in various
reduction techniques that are now tailor-made to the network setting, and
a pathway towards the classification of generic bifurcations in these
systems. Hidden Symmetry in Network Dynamical Systems | 11 October 2022 |

Dmitrii Mints (HSE Nizhny Novgorod) | **Abstract:** We consider regular Denjoy type homeomorphisms of the two-dimensional torus which are the most natural generalization of Denjoy homeomorphisms of the circle. In particular, they arise as Poincare maps induced on global cross-sections by leaves of one-dimensional orientable unstable foliations of some partially hyperbolic diffeomorphisms of closed three-dimensional manifolds. The non-wandering set of each regular Denjoy type homeomorphism is Sierpinski set and each such homeomorphism by definition is semiconjugate to the minimal translation on the two-dimensional torus. We introduce the complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is characterized by the minimial translation being semiconjugation of the given regular Denjoy type homeomorphism and by the countable set which is the union of some orbits of this translation. On topological classification of regular Denjoy type homeomorphisms | 22 March 2022 |

Snir Ben Ovadia (Pennsylvania State University) | **Abstract:** We introduce a class of orbits which may have zero Lyapunov exponents. We prove the existence of (weak) stable and unstable leaves for such orbits, and continue to prove the absolute continuity of these (weak) foliations. Using a shadowing lemma, we prove that every system whose Lyapunov exponents of hyperbolic orbits (on a transitive component) are not bounded from below admits such strictly weak foliations. Finally, we mention a few examples. Summable Orbits and Weak Foliations | 15 March 2022 |

Selim Ghazouani (Imperial College London) | **Abstract:** We will discuss a general programme, in the spirit of Palis' conjectures, for 1-dimensional dynamical systems deriving from flows on surfaces. It bears some ressemblances with the case of unimodal maps, with notable differences: absence of chaos and non-uniform hyperbolicity, and more complicated bifurcation loci and renormalisation theory. Global picture for the dynamics of foliations on surfaces (Part 2) | 1 March 2022 |

Selim Ghazouani (Imperial College London) | **Abstract:** We will discuss a general programme, in the spirit of Palis' conjectures, for 1-dimensional dynamical systems deriving from flows on surfaces. It bears some ressemblances with the case of unimodal maps, with notable differences: absence of chaos and non-uniform hyperbolicity, and more complicated bifurcation loci and renormalisation theory. Global picture for the dynamics of foliations on surfaces (Part 1) | 22 February 2022 |

Alexey Okunev (Loughborough University) | **Abstract:** There are two major obstacles to applying the averaging method, resonances and separatrices. We study averaging method for the simplest situation where both these obstacles are present at the same time, time-periodic perturbations of one-frequency Hamiltonian systems with separatrices. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so slow-fast Hamiltonian systems with two and a half degrees of freedom are included in our class).
Solutions passing through a resonance exhibit a small quasi-random jump, this is called scattering on a resonance. Some solutions passing through a resonance can also be captured into resonance, remaining near the resonance for a long time, but this only happens for small measure of initial data. Far from separatrices there are only finitely many resonances such that capture is possible, however, such resonances can accumulate on separatrices. We estimate how the amplitude of scattering on resonances and the measure of initial data captured into resonances decrease for resonances near separatrices. We also show that the infinite number of resonances near separatrices such that capture is possible can be split into a finite number of series such that dynamics near all resonances in the same series is close to each other and can be written in terms of the Melnikov function. We obtain realistic estimates on the accuracy of averaging method and on the measure of initial data badly described by averaging method (such as initial data captured into resonances) for solutions crossing separatrices. We also prove formulas for probability of capture into different domains after separatrix crossing. Our results can also be applied to perturbations of generic two-frequency integrable systems near separatrices, as they can be reduced to periodic perturbations of one-frequency systems.
Averaging and passage through resonances in two-frequency systems near separatrices | 25 January 2022 |

Felix Lequen (Laboratoire AGM – CY Cergy Paris Universite) | **Abstract:** The possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk we will be interested in the properties of these measures — in particular, whether they are absolutely continuous. First, I will quickly describe the case of Bernoulli convolutions, which can be seen as generalisations of the Cantor middle third set, and then the case of random iterations of matrices in SL(2, R) acting on the real projective line, where the stationary measure is unique under certain conditions, and is called the Furstenberg measure. It had been conjectured that the Furstenberg measure is always singular when the random walk has a finite support. There have been several counter-examples, and the aim of the talk will be to describe that of Bourgain, where the measure even has a very regular density. We will explain why the construction works for any simple Lie group, using the work of Boutonnet, Ioana, and Salehi Golsefidy on local spectral gaps in simple Lie groups. Bourgain's construction of finitely supported measures with regular Furstenberg measure | 18 January 2022 |

Emilio Corso (ETH Zurich) | **Abstract:** Distributional limit theorems for time averages along orbits of a discrete system or a flow are dynamical counterparts of the classical central limit theorem for empirical means of independent random variables. The talk will start out by introducing the complementary notions of spatial and temporal distributional limit theorem, in which the source of randomness is given, respectively, by space and time. It then aims to survey the state of the art regarding limit theorems for geodesic and horocycle flows on the unit tangent bundle of compact hyperbolic surfaces. Specifically, we shall place emphasis on our recent, streamlined rediscovery of the temporal limit theorem of Dolgopyat and Sarig along horocycle orbits; the proof relies, somewhat surprisingly at first glance, on the spatial limit theorem along geodesic orbits established by Sinai over half a century ago. Distributional limit theorems for geodesic and horocycle flows on compact hyperbolic surfaces | 14 December 2021 |

Isaia
Nisoli (Universidade Federal de Rio de Janeiro) | **Abstract:** In this talk I will introduce the notion of Noise Induced Order, a phenomenon by which the chaotic regime
of a deterministic system is destroyed in the presence of noise, quantitatively measured through a transition of the top Lyapunov
exponent from positive to negative as the noise amplitude increases.
This phenomenon is strictly connected with non-uniform hyperbolicity; in [2] this phenomenon was proved for a family of unimodal
maps with a critical point of order bigger than 2.67835.
Today I will introduce the results of [1], where we prove the existence of this phenomenon for skew products with a non-uniformly expanding base. This allows us to prove Noise-Induced Order for the Contracting Lorenz 2-dimensional map.
[1] Blumenthal A., Nisoli I. "Noise Induced Order for Skew-Products over a Non-Uniformly Expanding Base" preprint, arXiv: 2109.12183
[2] Nisoli I. ”How does noise induce order?” preprint, arXiv:2003.08422
Noise Induced Order for Skew-Products over a Non-Uniformly Expanding Base | 8 December 2021 |

Andrey Morozov (HSE Nizhny Novgorod) | **Abstract:** According to Thurston's classification, the set of homotopy classes of homeomorphisms defined on closed orientable surfaces of negative curvature is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence in it of a homeomorphism called the Thurston canonical form and which is exactly one of the following types, respectively: a periodic homeomorphism, reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a algebraically finite order homeomorphism, a pseudo-anosov homeomorphism. Thurston's canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In each homotopy class from the first subset A.N. Bezdezhnykh and V.Z. Grines constructed a gradient-like diffeomorphism. R.V. Plykin and A. Yu. Zhirov announced a method for constructing a structurally stable diffeomorphism in each homotopy class from the fourth subset. The non-wandering set of this diffeomorphism consists of a finite number of source orbits and a single one-dimensional attractor. In this paper, we describe the construction of a structurally stable diffeomorphism in each homotopy class from the second subset. The constructed representative is a Morse-Smale diffeomorphism with an orientable heteroclinic intersection. Realization of homeomorphisms of surfaces of algebraically finite order by Morse-Smale diffeomorphisms with orientable heteroclinic intersection | 30 November 2021 |

Sergey Zelik (University of Surrey) | **Abstract:** We will discuss the further regularity of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than C1 plus epsilon-regularity for such manifolds (for some positive, but small epsilon). Nevertheless, as we will see, under the natural assumptions, the obstacles to the existence of a Cn-smooth inertial manifold (where n is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the C1 plus epsilon-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.
Smooth extensions for Inertial Manifolds of semilinear parabolic equations | 23 November 2021 |

Danila Shubin (HSE Nizhny Novgorod) | **Abstract:** In this talk, we will consider the topological properties of three-dimensional manifolds that admit Morse-Smale flows without fixed points (nonsingular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be established more accurately and refined in the case of a small number of orbits. For example, in the case of a flow with two nontwisted (having a tubular neighborhood homeomorphic to a filled torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an HMS flow with two orbits is a lens space. Previously, it was believed that all simple manifolds admitting NMS flows with at most three non-twisted orbits have the same topology. It turns out that the ambient manifolds of suspensions over Morse-Smale diffeomorphisms with three periodic orbits are not lens spaces and, moreover, are pairwise non-homeomorphic. It follows from the results of this paper that there is a countable set of pairwise non homeomorphic three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an HMS flow with at most three orbits is a lens space. Topology of ambient manifolds of nonsingular flows with three nontwisted orbits | 16 November 2021 |

Deniz Eroglu | | 29 June 2021 |

Sajjad Bakrani | | 29 June 2021 |

Ralf Tönjes | **Abstract:** Complex networks are abundant in nature and many share an important
structural property: they contain a few nodes that are abnormally
highly connected (hubs). Some of these hubs are called influencers
because they couple strongly to the network and play fundamental
dynamical and structural roles. Strikingly, despite the abundance of
networks with influencers, little is known about their response to
stochastic forcing. Here, for oscillatory dynamics on influencer
networks, we show that subjecting influencers to an optimal intensity
of noise can result in enhanced network synchronization. This new
network dynamical effect, which we call coherence resonance in
influencer networks, emerges from a synergy between network structure
and stochasticity and is highly nonlinear, vanishing when the noise is
too weak or too strong. Our results reveal that the influencer
backbone can sharply increase the dynamical response in complex
systems of coupled oscillators. Coherence resonance in influencer networks
| 22 June 2021 |

Lars Grüne | **Abstract:** Abstract: Optimal Feedback Control is a computationally intensive task. The classical approach to a numerical solution of this problem via the Hamilton-Jacobi-Bellman PDE in continuous time or the Bellman equation in discrete time may fail already for very low dimensional systems because of its high computational complexity.
In contrast to this, modern optimisation based control methods like model predictive control or deep reinforcement learning are able to solve such problems approximately with much larger state dimension. In this talk we argue that this is possible because the dynamics of the problem exhibits some kind of redundancy, which can efficiently be exploited by these algorithms. We will explain which kind of redundancy this can be and why it helps to reduce the computational complexity of these methods. Using Redundancy of the Dynamics in Nonlinear Optimal Feedback Control
| 15 June 2021 |

Hedy Attouch (Université Montpellier, France) | **Abstract:** \begin{abstract}
We report on recent advances regarding the acceleration of first-order algorithms for continuous optimization. We rely on the damped inertial dynamics driven by the gradient of the function $ f $ to be minimized, and on the algorithms obtained by temporal discretization. The first (main) part of the lecture is devoted to convex optimization in a general Hilbert framework. We review classical results, from Polyak's heavy ball with friction method to Nesterov's accelerated gradient method.
Then we introduce the geometric damping driven by the Hessian which intervenes in the dynamic in the form $\nabla^2 f (x(t)) \dot{x} (t)$. By treating this term as the time derivative of $ \nabla f (x (t)) $, this gives, in discretized form, first-order algorithms [1].
This geometric damping makes it possible to attenuate the oscillations.
Besides the fast convergence of the values, the algorithms thus obtained show a rapid convergence towards zero of the gradients.
Numerical results for structured optimization and Lasso problems support our theoretical results.
Next, we consider the introduction into the dynamics/algorithms of a Tikhonov regularization term with asymptotic vanishing coefficient.
Based on a proper tuning of the parameters, we obtain both fast convergence of the values and strong convergence towards the minimum norm solution [2]. Then, for linear constrained convex optimization, we introduce a new dynamic approach to the inertial ADMM algorithms, and thus obtain optimal convergence rates.
We finally describe some basic tools to deal with inertial algorithms for non-convex non-smooth optimization:
quasi-gradient dynamics, Kurdyka-Lojasewicz inequality, tame analysis and semi algebraic functions [3].
%
\end{abstract}
\medskip
\begin{footnotesize}
\noindent [1] \; H. Attouch, Z. Chbani, J. Fadili, H. Riahi, First-order optimization algorithms via inertial systems with Hessian driven damping, Math. Program., (2020)\\
https://doi.org/10.1007/s10107-020-01591-1, arXiv:1907.10536v2 [math.OC]
\smallskip
\noindent [2] \; H. Attouch, S. Laszlo, Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution, (2021),
arXiv:2104.11987 [math.OC]
\smallskip
\noindent [3] \; H. Attouch, R.I. Bo\c t, E.R. Csetnek,
Fast optimization via inertial dynamics with closed-loop damping,
Journal of the European Mathematical Society, (2021),\\ arXiv:2008.02261v3 [math.OC]
\end{footnotesize} Acceleration of first-order optimization algorithms via damped inertial dynamics | 8 June 2021 |

Michael Muehlebach | **Abstract:** My talk will highlight connections between dynamical systems and optimization and present an analysis of accelerated first-order optimization algorithms. I will show how the continuous dependence of iterates with respect to their initial condition can be exploited for characterizing the convergence rate. The result establishes criteria for accelerated convergence of any momentum-based optimization algorithm, which are easily verifiable. The criteria are necessary and sufficient and therefore precisely characterize optimization algorithms that are accelerated. The analysis applies to non-convex functions, unifies discrete-time and continuous-time models, and rigorously explains why structure-preserving discretization schemes are important in optimization.
If time permits, I will further discuss lower bounds on the complexity of gradient-based algorithms from a continuous-time point of view. A normalization of time will make the discussion of continuous-time convergence rates meaningful and recovers the well-known bounds from the discrete-time setting. Furthermore, the talk will highlight algorithms that achieve the proposed lower bounds, even when the function class under consideration includes certain non-convex functions. Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives | 8 June 2021 |

Qianxiao Li | **Abstract:** Abstract: In this talk, I will discuss some recent approximation results on the intersection of dynamical systems and machine learning. In the first direction, I will present some universal approximation theorems for continuous-time idealizations of deep residual networks. In the second, I will focus on the approximation of dynamical relationships by recurrent and convolutional structures for time series, whose analysis reveals some interesting connections between approximation, memory and sparsity. Approximation theory for machine learning and dynamical systems
| 1 June 2021 |

Soon Hoe Lim | **Abstract:** We provide a general framework for studying recurrent neural networks (RNNs) trained by injecting noise into hidden states. Specifically, we consider RNNs that can be viewed as discretizations of stochastic differential equations driven by input data. This framework allows us to study the implicit regularization effect of general noise injection schemes by deriving an approximate explicit regularizer in the small noise regime. We find that, under reasonable assumptions, this implicit regularization promotes flatter minima; it biases towards models with more stable dynamics; and, in classification tasks, it favors models with larger classification margin. Sufficient conditions for global stability are obtained, highlighting the phenomenon of stochastic stabilization, where noise injection can improve stability during training. Our theory is supported by empirical results which demonstrate improved robustness with respect to various input perturbations, while maintaining state-of-the-art performance. Noisy Recurrent Neural Networks
| 1 June 2021 |

Manjunath Gandhi | **Abstract:** The talk is about finding models from data assuming the temporal data was generated from a dynamical system. Data-driven methods employing the Koopman operator, and algorithms in reservoir computing transform data into another phase space for model construction. They have shown great promise in forecasting some chaotic dynamical systems. Finding the right observables for the Koopman operator in data-driven approaches is an open problem -- existing methods like (SINDy) determine a set of observables only after assuming that the vector field to be learned lies in the span of a set of predetermined functions. Igor Mezi? in a Fields symposium talk last year mentioned an ambitious idea of finite faithful representations. This is to obtain observables that determine a topologically conjugate system like in Takens delay embedding. Echo state networks (ECNs) employing the reservoir computing paradigm transform the temporal data as a nonautonomous attractor in another space, but there is no guarantee of the existence of a learnable map. We solve both problems through an idea of causal embedding a notion of embedding temporal data into another space and consider its implementation through recurrent conjugate networks (RCNs), an adaptation of ESNs. RCNs render a learnable map that is topological conjugate to the system generating the data (as in Takens delay embedding), and also determines observables forming a finite faithful representation for the Koopman operator. RCNs give exceptional long-term consistency in numerical forecasting experiments as well.
This is a joint work with Adriaan de Clrecq. Universal set of Observables for Koopman's Operator through Causal Embedding | 25 May 2021 |

John Harlim (Pennsylvania State University) | **Abstract:** Abstract: In the talk, I will discuss a general closure framework to compensate for the model error arising from missing dynamical systems. The proposed framework reformulates the model error problem into a supervised learning task to estimate a very high-dimensional closure model, deduced from the Mori-Zwanzig representation of a projected dynamical system with projection operator chosen based on Takens embedding theory. Besides theoretical convergence, this connection provides a systematic framework for closure modeling using available machine learning algorithms. I will demonstrate numerical results using a kernel-based linear estimator as well as neural network-based nonlinear estimators. If time permits, I will also discuss error bounds and mathematical conditions that allow for the estimated model to reproduce the underlying stationary statistics, such as one-point statistical moments and auto-correlation functions, in the context of learning Ito diffusions. Machine learning of missing dynamical systems | 18 May 2021 |

Daniel Wilczak (Jagiellonian University, Poland) | **Abstract:** Abstract:
I will present an algorithm for rigorous computation of Poincar\'e maps. It addresses the following questions:
What is the optimal choice (in the sense of obtained enclosures) of a Poincar\'e section near a periodic orbit?
If the sections are fixed (like guards in hybrid systems), can we reduce overestimation by appropriate choice of coordinates on these sections?
Can we take advantage from the knowledge on internal representation of the solution to IVP in a rigorous ODE solver to reduce overestimation in computation of Poincar\'e maps?
The algorithm [3] is already implemented and freely available as a part of the CAPD library [1,2]. Theoretical considerations will be supported by numerical tests [3].
References:
[1] CAPD::DynSys Computer Assisted Proofs in Dynamics, a C++ package for rigorous numerics. http://capd.ii.uj.edu.pl/
[2] T. Kapela, M. Mrozek, D. Wilczak, P Zgliczy?ski, {CAPD::DynSys:} a flexible {C}++ toolbox for rigorous numerical analysis of dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 10.1016/j.cnsns.2020.105578
[3] T. Kapela, D. Wilczak, P Zgliczy?ski, Recent advances in rigorous computation of Poincar\'e maps, Communications in Nonlinear Science and Numerical Simulation, in review Recent advances in rigorous computation of Poincaré maps
| 11 May 2021 |

Gary Froyland (University of New South Wales) | **Abstract:** In the first part of the talk I will describe how the spectrum and eigenfunctions of transfer operators can extract cycles from possibly high-dimensional spatio-temporal information. This will be illustrated by extracting a canonical cycle for the El-Nino Southern Oscillation (ENSO) from sea-surface temperature data. We are able to produce a "rectified" cycle that provides more detail in rapid transitions.
In the second part of the talk I will introduce new theory to handle the birth and death of coherent sets. Coherent sets are regions of phase space that minimally deform and mix under general aperiodic advection. They are by definition material, evolving with the underlying flow or dynamics. Methods for identifying coherent sets, and Lagrangian coherent structures more generally, rely on coherence being present throughout a specified time interval. In reality, coherent structures are ephemeral, continually appearing and disappearing. I will present a new construction, based on the dynamic Laplacian, that relaxes this materiality requirement in a natural way, and provides the means to resolve the births, lifetimes, and deaths of coherent structures.
Extracting the ENSO cycle from observations and the birth and death of coherent sets | 4 May 2021 |

Alexandre Mauroy | **Abstract:** The Koopman operator framework provides a linear description of nonlinear systems, which can be leveraged for data-driven applications. This Koopman operator-based approach will be reviewed in this talk, with a focus on finite-dimensional approximations of the operator used for specific data-driven applications. We will first consider the problem of nonlinear identification and parameter estimation, exploiting the key idea that identifying (an approximation of) the linear Koopman operator is equivalent to identifying the nonlinear underlying dynamics. Two dual linear techniques will be presented, which are complemented with convergence properties and also extended to the identification of nonlinear PDEs. In the last part of the talk, we will introduce a novel approximation of the operator based on reservoir computing, with potential applications to prediction and spectral analysis. Data-driven Koopman operator-based methods
| 27 April 2021 |

Felix Dietrich | On the Koopman Operator of Algorithms
| 27 April 2021 |

Woojin Kim (Duke University) | **Abstract:** This talk introduces a method for characterizing the dynamics of
time-evolving data within the framework of topological data analysis
(TDA), specifically through the lens of persistent homology. Popular
instances of time-evolving data include flocking or swarming behaviors
in animals, and social networks in the human sphere. A natural
mathematical model for such collective behaviors is that of a dynamic
metric space. In this talk I will describe how to extend the well-known
Vietoris-Rips filtration for metric spaces to the setting of dynamic
metric spaces. Also, we extend a celebrated stability theorem on
persistent homology for metric spaces to multiparameter persistent
homology for dynamic metric spaces. In order to address this stability
property, we extend the notion of Gromov-Hausdorff distance between
metric spaces to dynamic metric spaces. This talk will not require any
prior knowledge of TDA. This talk is based on joint work with Facundo
Memoli and Nate Clause. The Persistent Topology of Dynamic Data | 23 March 2021 |

Christoph Kawan (University of Munich) | **Abstract:** Networked control systems violate standard assumptions of classical control theory.
One of the many challenges in their analysis and design concerns information constraints
present in the communication between sensors, controllers and actuators. A fundamental
question in this field is thus concerned with the smallest rate of information flowing
from the sensors to the controller, above which a given control task can be solved.
In this talk, we address this question in the context of set-stabilization for nonlinear
systems. To obtain exact results, we impose the assumption of uniform hyperbolicity on
the sets under consideration, which provides us with powerful tools such as shadowing
and hyperbolic volume estimates. As we can then see, the minimal information rate is
closely related to quantities extensively studied in smooth dynamical systems: escape
rates, topological pressure, measure-theoretic entropy and Lyapunov exponents. Control of chaos with minimal information transfer | 23 March 2021 |

Yoshito Hirata (University of Tsukuba) | **Abstract:** Distinguishing deterministic systems from stochastic systems has been
discussed for a long time. But, such analysis had been qualitative or
contrasting nonlinear deterministic systems with linear stochastic
systems. Thus, we could not identify nonlinear stochastic systems with
some hypothesis tests. Here, we propose to use permutations or
recurrence plots for distinguishing stochastic systems from
deterministic systems with a hypothesis test. Therefore, permutations
and recurrence plots can be used also for analyzing a time series
generated from nonlinear stochastic systems. Unified time series analysis for nonlinear deterministic/stochastic systems
| 16 March 2021 |

Erik Bollt (Clarkson University) | **Abstract:** In the spirit of optimal approximation and reduced order modelling the goal of DMD methods and variants is to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. That Koopman eigenfunctions follow a linear PDE that is solvable by the method of characteristics yields several interesting relationships between geometric and algebraic properties. We focus on contrasting cardinality, algebraic multiplicity and other geometric aspects with the introduction of an equivalence class, “primary eigenfunctions,” for those eigenfunctions with identical sets of level sets. We present a construction that leads to functions on the data surface that yield optimal Koopman eigenfunction DMD, (oKEEDMD). We will also describe that disparate systems can be “matched” transformed by a diffeomorphism constructed via eigenfunctions from each system, a reinterpretation of integrability, computationally stated by our “matching extended dynamic mode decomposition (EDMD)” (EDMD-M). Geometry and Good Dictionaries for Koopman Analysis of Dynamical Systems
| 9 March 2021 |

Sigurður Freyr Hafstein (University of Iceland
) | **Abstract:** Attractors and their basins of attraction in deterministic dynamical systems are most commonly studied using the Lyapunov stability theory.
Its centerpiece is the Lyapunov function, which is an energy-like function from the state-space that is decreasing along all solution trajectories.
The Lyapunov stability theory for stochastic differential equations is much less developed and, in particular,
numerical methods for the construction of Lyapunov functions for such systems are few and far between.
We discuss the general problem and present some novel numerical methods. Lyapunov functions for stochastic differential equations: Theory and computation | 2 March 2021 |

Josef Teichmann (ETH) | **Abstract:** We show that general stochastic differential equations (SDEs)
driven by Brownian motions or jump processes can be approximated by
certain SDEs with random characteristics in the spirit of reservoir
computing. Notions of semi-martingale signatures and its randomized
versions are applied here. Reservoir Computing for SDEs | 23 February 2021 |

Kathrin Padberg-Gehle (Leuphana University of Lüneburg) | **Abstract:** Transport and mixing processes in fluid flows are crucially influenced by coherent structures and the
characterisation of these Lagrangian objects is a topic of intense current research. While established
mathematical approaches such as variational or transfer operator based schemes require full knowledge
of the flow field or at least high resolution trajectory data, this information may not be available in applications.
In this talk, we review different spatio-temporal clustering approaches and show how these can be used to
identify coherent behaviour in flows directly from Lagrangian trajectory data. We demonstrate the applicability
of these methods in a number of example systems, including geophysical flows and turbulent convection. Data-based analysis of Lagrangian transport
| 9 February 2021 |

Armen Shirikyan (Université de Cergy-Pontoise) | **Abstract:** A well-known fact from the theory of stochastic differential equations on a compact manifold is that the validity of Hörmander's condition for the vector fields implies the uniqueness of a stationary measure and its exponential stability in the total variation metric. We study this problem in an abstract setting for a Markovian random dynamical system in a compact metric space. It is proved that the global controllability to a point and solid controllability from that point imply the uniqueness and exponential mixing of a stationary measure, provided that the noise satisfies a mild non-degeneracy hypothesis. The result is illustrated on a differential equation with random coefficients on a compact manifold. We shall also discuss briefly a generalisation of the abstract result that is applicable to a class of randomly forced PDEs. A simple mixing criterion for random dynamical systems and an application | 2 February 2021 |

Andrei Agrachev (SISSA, Trieste) | **Abstract:** Abstract: Given a control system on a smooth manifold M, any admissible control function generates a flow, i.e. a one-parametric family of diffeomorphisms of M. We give a sufficient condition for the system that guarantees the existence of an arbitrary good uniform approximation of any isotopic to the identity diffeomorphism by an admissible diffeomorphism and provide simple examples of control systems on \mathbb R^n, \mathbb T^n and \mathbb S^2 that satisfy this condition. This work is a joint work with A. Sarychev (Florence) motivated by the deep learning of artificial neural networks treated as a kind of interpolation technique. Control of Diffeomorphisms with Applications to Deep Learning | 26 January 2021 |

Marian
Mrozek (Uniwersytet Jagiellonski
) | **Abstract:** The ease of collecting enormous amounts of data in the present world
together with problems in gaining useful knowledge out of it
stimulate the development of mathematical tools to deal with the situation.
In particular, in the case of data collected from dynamic processes,
the methods developed for computer assisted proofs in dynamics
have been adapted to the new challenges. Closely related to the techniques
of multivalued maps used in computer assisted proofs in dynamics
are the methods centered around the concept of combinatorial vector field,
a concept introduced twenty years ago by R. Forman.
In the talk I will review some recent results concerning topological
invariants for combinatorial vector fields and their extensions. Topological Methods in Combinatorial Dynamics | 19 January 2021 |

Peter Giesl (University of Sussex) | **Abstract:** Abstract: A contraction metric is a Riemannian metric, with respect to which the distance between adjacent solutions of an ordinary differential equation (ODE) decreases.
A contraction metric can be used to prove existence and uniqueness of an equilibrium of an autonomous ODE and determine a subset of its basin of attraction without requiring information about its location. Moreover, a contraction metric is robust to small perturbations of the system.
We will prove a converse theorem, showing the existence of a contraction metric for an equilibrium by characterising it as a matrix-valued solution of a certain linear partial differential equation (PDE). This leads to a construction method by numerically solving the matrix-valued PDE using mesh-free collocation. We use and present a recent extension of mesh-free collocation of scalar-valued functions, solving linear PDEs, to matrix-valued ones. Finally, we briefly discuss a method to verify that the computed metric satisfies the conditions of a contraction metric.
This is partly work with Holger Wendland, Bayreuth as well as Sigurdur Hafstein and Iman Mehrabinezhad, Iceland. Existence and construction of a contraction metric as solution of a matrix-valued PDE | 12 January 2021 |

Alex Blumenthal (Georgia Tech) | **Abstract:**
The Lyapunov exponent measures the rate at which nearby initial conditions of a dynamical system converge or diverge: a positive exponent, indicating divergence, is a classical hallmark of chaotic behavior. Unfortunately, it can be profoundly difficult to estimate the Lyapunov exponents of dynamical systems of physical interest. On the other hand, for volume-preserving systems subjected to random noise, positivity of the Lyapunov exponent must hold unless some severe degenerate behavior is present (due to work of Furstenberg and many others). A significant drawback of this theory is that it provides no quantitative estimate of Lyapunov exponents, only positivity. In this talk, I will discuss recent work with J. Bedrossian and S. Punshon-Smith providing a new, more quantitative perspective on Furstenberg-type criteria for Lyapunov exponents for SDE using a Fisher information-type identity and a quantitative re-working of aspects of Hormander’s hypoellipticity theory. As an application, we are able to establish positivity of the Lyapunov exponent for a class of systems subjected to weak dissipation effects (e.g., drag, viscosity), including the Lorenz 96 system with arbitrarily many oscillators.
A Fisher information perspective on Lyapunov exponents with applications to the stochastically-driven Lorenz 96 model | 8 December 2020 |

Michael Scheutzow (TU Berlin) | **Abstract:** We discuss the change of stability behaviour of a deterministic dynamical systems in Euclidean space under the addition of white noise. It is known that noise can have a stabilizing or destabilizing effect depending on the
underlying system. We provide an example of a dynamical system in the plain which exhibits blow-up in finite time for almost all initial conditions such that for additive noise of arbitrarily small intensity the system
has strong stability properties: it is not only stable in the sense that it does not blow-up but it even admits a random set attractor.
This is joint work with Matti Leimbach (Berlin) and Jonathan Mattingly (Duke, Durham). More elements at https://arxiv.org/abs/2009.10573
Noise-induced strong stabilization | 1 December 2020 |

Ard Louis (University of Oxford) | **Abstract:** One of the most surprising properties of deep neural networks (DNNs) is that they perform best in the overparameterized regime. We are all taught in a basic statistics class that having more parameters than data points is a terrible idea. This intuition can be formalised in standard learning theory approaches, based for example on model capacity, which also predict that DNNs should heavily over-fit in this regime, and therefore not generalise at all. So why do DNNs work so well in a regime where theory says they should fail?
A popular strategy in the literature has been to look for some dynamical property of stochastic gradient descent (SGD) acting on a non-convex loss-landscape in order to explain the bias towards functions with good generalisation. Here I will present a different argument, namely that DNNs are implicitly biased towards simple (low Kolmogorov complexity) solutions at initialisation [1]. This Occam's razor like effect fundamentally arises from a version of the coding theorem of algorithmic information theory, applied to input-output maps [2]. We also show that for DNNs in the chaotic regime, the bias can be tuned away, and the good generalisation disappears. For highly biased loss-landscapes, SGD converges to functions with a probability that can, to first order, be approximated by the probability at initialisation [3]. Thus, even though, to second order, tweaking optimisation hyperparameters can improve performance, SGD itself does not explain why DNNs generalise well in the overparameterized regime. Instead it is the intrinsic bias towards simple (low Kolmogorov complexity) functions that explains why they do not overfit. Finally, this function based picture allows us to derive rigorous PAC-Bayes bounds that closely track DNN learning curves and can be used to rationalise differences in performance across architectures.
[1] Deep learning generalizes because the parameter-function map is biased towards simple functions, Guillermo Valle Pérez, Chico Q. Camargo, Ard A. Louis arxiv:1805.08522
[2] Input–output maps are strongly biased towards simple outputs, K. Dingle, C. Q. Camargo and A. A. Louis Nature Comm. 9, 761 (2018)
[3] Is SGD a Bayesian sampler? Well, almost, Chris Mingard, Guillermo Valle-Pérez, Joar Skalse, Ard A. Louis arxiv:2006.15191
Kolmogorov complexity and explaining why neural networks generalise so well (using a function based picture) | 24 November 2020 |

Soon Hoe Lim (Nordita, KTH Royal Institute of Technology and Stockholm University) | **Abstract:** Critical transitions are widespread in many systems in nature. Often times these transition events are rare and are induced by noise taking the form of a fast driving signal. Since such events could lead to significant effects, it is important to develop effective methods to predict them. In this talk, we will discuss the problem of predicting rare noise-induced critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster chaotic process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. Predicting noise-induced critical transitions in multiscale dynamical systems | 17 November 2020 |

Yuka Hashimoto (Keio University
) | **Abstract:** RKHM (Reproducing kernel Hilbert C*-module) is a generalization of RKHS (Reproducing kernel Hilbert space) which is characterized by a C*-algebra-valued positive definite kernel and inner product associated with this kernel. Regarding RKHS, it has been actively researched for data analysis. Moreover, time-series data analysis by Perron-Frobenius and Koopman operators in RKHSs has been investigated. In this framework, the time-series data is assumed to be generated from a dynamical system and we can estimate Perron-Frobenius and Koopman operators only by the data. Since these operators are linear, we can apply the theory of linear algebra for the estimation. However, for interacting dynamical systems, information about interactions tend to degenerate in RKHSs and extracting such information from given data is difficult. Therefore, we consider using RKHMs instead of RKHSs for interacting dynamical systems. Since inner products in RKHMs are C*-algebra-valued, they capture more information about interactions than complex-valued ones. As a result, we can extract information about interactions from given data.
Time-series data analysis with Reproducing kernel Hilbert C*-modules | 10 November 2020 |

Enrique Zuazua (1.) FAU-AvH, Erlangen, Germany,
2.) Fundación Deusto, Bilbao, Basque Country, Spain,
3.) Universidad Autónoma de Madrid, Spain) | **Abstract:** We analyse reaction-diffusion models arising in social and biological sciences. Frequently, they play an important role in applications, when avoiding population extinction or propagation of infectious diseases, enhancing multicultural features, etc. But, when doing that, often, the mathematical formulation of these issues requires of the concepts and tools from Control Theory, which may be reinterpreted in terms of the dynamical system behaviour of Residual Neural Networks.
In this talk, oriented to a broad audience, and avoiding unnecessary technical difficulties, we should explain how these areas of Applied Mathematics meet, pointing some challenging problems for future research.
Reaction-diffusion in Social and Biological Sciences: Dynamics and ResNet control | 3 November 2020 |

Nils Berglund (University of Orleans) | **Abstract:** Abstract: We consider stochastic differential equations describing the motion of an overdamped Brownian particle in a periodically oscillating double-well potential. Our main objective is to describe the distribution of
transition times between potential wells. Without periodic perturbation, the answer to this question is well known: the distribution of transition times is asymptotically exponential, with an expectation given by the so-called Eyring-Kramers law. With the periodic perturbation, the equation becomes non-reversible, which makes
the analysis much harder. We will present results both on the shape of the distribution of transition times, and on its expectation for a range of oscillation frequencies. Partly based on joint work with Barbara Gentz (Bielefeld). Precise estimates on noise-induced transitions between oscillating double-well potentials | 27 October 2020 |

Konstantin Mischaikow (Rutgers University) | **Abstract:** Using examples from biology and data driven science I will argue that we need a new framework in which to discuss nonlinear dynamics. I will propose such a framework that takes the form of combinatorics, order theory, and algebraic topology.
I will discuss how this new framework relates to classical dynamics, show that the combinatorial approach allows for extremely efficient computation, and suggest future directions of research. Dynamic Clades: A coarse approach to nonlinear dynamics
| 20 October 2020 |

Peter Koltai (FU Berlin) | **Abstract:** The decomposition of the state space of a dynamical system into almost invariant sets is important for understanding its essential macroscopic behavior. The concept is reasonably well understood for autonomous dynamical systems, and recently a generalization appeared for non-autonomous systems: coherent sets. Aiming at a unified theory, in this talk we will first present connections between the measure-theoretic autonomous and non-autonomous concepts. We shall do this by considering the augmented state space. Second, we will extend the framework to finite-time systems, and show that it is especially well-suited for manipulating the mixing properties of the dynamics. Coarse-graining and manipulation of transport in non-autonomous systems | 13 October 2020 |

Stefan Klus (University of Surrey) | **Abstract:** Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. In this talk, we present a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. Under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This enables us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. More details at https://arxiv.org/abs/2005.13231 Kernel-based approximation of the Koopman generator and Schrödinger operator | 6 October 2020 |

Sergey Zelik (University of Surrey) | **Abstract:** We discuss the dynamics generated by
weakly damped wave equations in bounded 3D domains where the damping
exponent depends explicitly on time and may change sign. It is shown that
in the case when the non-linearity is superlinear, the considered equation
remains dissipative if the weighted mean value of the dissipation rate
remains positive.
Two principally different cases are considered. In the case when this mean is uniform (which corresponds to deterministic dissipation rate), it is shown that the considered system possesses smooth
uniform attractors as well as non-autonomous exponential attractors.
In the case where the mean is not uniform (which corresponds to the random
dissipation rate), the tempered random attractor is constructed. In contrast to the usual situation, this random attractor is expected to have
infinite dimension. The simplified model example which demonstrates infinite-dimensionality of the random attractor is also presented. Deterministic and random attractors for a wave equation with sign changing damping | 28 July 2020 |

Dan Rust (Open University) | **Abstract:** Quasicrystals are non-periodic arrangements of atoms which nevertheless
possess a huge amount of long-range order. They have interested
dynamicists since their discovery in the late 20th century, as one can
assign a dynamical system to a quasicrystal which encodes structural
properties of the crystal. In this way, questions like 'How many
different constellations of atoms of a particular size appear in the
crystal?' are translated into dynamical questions such as 'What is the
entropy of the dynamical system?'. In general, long-range order and
positive entropy do not marry up well with one another, and so it's
interesting that a class of quasicrystals generated by 'random
substitutions' display both simultaneously. I will give a brief overview
of this topic and spend some time introducing random substitutions and
their dynamical properties. In particular, we'll see that the
'eigenvalues' of a random substitution encode mixing properties of the
dynamical system. We'll also see that we can assign a 'Rauzy fractal' to
random substitutions which provides a suitable window for a
cut-and-project scheme whose model set has the quasicrystal embedded. Quasicrystals, hierarchies and entropy | 16 June 2020 |

Olga Lukina (University of Vienna) | **Abstract:** Let G be a countable group which acts on a Cantor set X effectively and equicontinuously. A point x in X has trivial holonomy if any element in the isotropy group of the action at x fixes an open neighborhood of x. It is well-known that the set of points with trivial holonomy is residual in the Cantor set, that is, points with trivial holonomy are topologically generic. In the talk, we give sufficient conditions on the action, under which points with trivial holonomy are generic in the measure-theoretical sense, with respect to the ergodic measure on X. We also discuss the applications of this results to the study of the structure of invariant random subgroups of G. This is joint work with Maik Gröger. Points without holonomy for group actions on Cantor sets | 9 June 2020 |

Toby Hall (University of Liverpool) | **Abstract:** I’ll present an overview of recent research on inverse limits of unimodal maps with Boyland and de Carvalho, including mode-locking of maximal itineraries; prime ends of Barge-Martin embeddings; and semi-conjugacy to sphere homeomorphisms. A common theme is the central role played by the height $q(f)\backslash in[0,1/2]$ of a unimodal map f. Inverse limits of unimodal maps | 2 June 2020 |

Vladislav Sidorenko (Keldysh Institute of Applied Mathematics) | **Abstract:** Considering the evolution of a weakly perturbed Keplerian motion under the scope of the restricted three-body problem M.L.Lidov (1961) and Y.Kozai (1962) discovered independently coupled oscillations of eccentricity and inclination (KL-cycles). Their classical investigations were based on the integrable model of the secular evolution obtained after double averaging of the disturbing function approximated by the first non-trivial term (more precisely, by the quadruple term) in the series expansion with respect to the ratio of semimajor axis of the disturbed body and the disturbing body.
If the next (octupole) term is kept in the expression of the disturbing function, then the longterm modulation of the KL-cycles can be established (Ford et al., 2000; Katz et al., 2011; Lithwick, Naoz, 2011). In particular, the flips become possible from prograde to retrograde orbit and back again. Since flips are observed only in the case of the disturbing body motion in the orbit with non-zero eccentricity, the term “Eccentric Kozai-Lidov Effect” (or EKL-effect) was proposed in (Lithwick, Naoz, 2011) to specify such a dynamical behavior.
We demonstrate that the EKL-effect can be interpreted as a resonance phenomenon. With this aim we write down the motion equations in terms of the “action-angle” variables provided by the integrable Kozai-Lidov model. It turns out that for some initial values the resonance is degenerate and the usual “pendulum” approximation is insufficient to describe the evolution of the resonance phase. The analysis of the related bifurcations allows us to estimate the typical time between the successive flips for different parts of the phase space. The Eccentric Kozai-Lidov Effect as a Resonance Phenomenon | 10 March 2020 |

Stergios Antonakoudis (Imperial College) | **Abstract:** Studying the existence of fixed points for holomorphic maps on Teichmüller spaces serves as a framework for proving 'geometrization' theorems, such as Thurston's topological characterisation of hyperbolic 3-manifolds. In this talk, we will prove a general theorem addressing the existence of fixed points for holomorphic maps on Teichmüller spaces and a more general class of complex domains by focusing on the intrinsic shape of these domains and leveraging a novel idea that combines arguments from (complex) geometry and elementary combinatorics. We'll also discuss a geometric generalisation and possible applications, obtained by a distillation of the ideas and arguments involved in the proofs.
Fixed point theorems for holomorphic maps on Teichmüller spaces and beyond. | 3 March 2020 |

Maximilian Engel (TU Munich) | **Abstract:** For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines isochrons as the cross-sections of the orbit with fixed return time under the flow, or, equivalently, as the stable manifolds foliating neighborhoods of the limit cycle. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator.
We introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to introduce a random version of isochron maps whose level sets coincide with the random stable manifolds. Furthermore, we sketch how this random dynamical systems interpretation may be linked to the physics approaches by appropriate averaging. A random dynamical systems perspective on isochronicity for stochastic oscillations | 25 February 2020 |

Feliks Nüske (Paderborn University) | **Abstract:** Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory -- with extended dynamic mode decomposition (EDMD) being a cornerstone of the field. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT) format -- have become a valuable tool for the solution of large-scale problems in a number of fields. In this work, we combine EDMD and the TT format, enabling the application of EDMD to high-dimensional problems in conjunction with a large set of features. We present the construction of different TT representations of tensor-structured data arrays. Furthermore, we also derive efficient algorithms to solve the EDMD eigenvalue problem based on those representations and to project the data into a low-dimensional representation defined by the eigenvectors. We prove that there is a physical interpretation of the procedure and demonstrate its capabilities by applying the method to benchmark data sets of molecular dynamics simulation. Tensor-based EDMD for the Koopman analysis of high-dimensional systems | 20 February 2020 |

Stefan Klus (Freie Universität Berlin) | **Abstract:** We present a data-driven method for the approximation of the
Koopman generator. It can be used for computing eigenvalues,
eigenfunctions, and modes of the generator and also for system
identification. In addition to learning the governing equations of
deterministic systems, which then reduces to SINDy (sparse
identification of nonlinear dynamics), it is possible to identify the
drift and diffusion terms of stochastic differential equations from
data. Moreover, it enables us to derive coarse-grained models of
high-dimensional systems and to determine efficient model predictive
control strategies. Data-driven approximation of the Koopman generator: Model
reduction, system identification, and control | 18 February 2020 |

James Waterman (University of Liverpool) | **Abstract:** The Hausdorff dimension of the Julia set of transcendental entire and meromorphic functions has been widely studied. We review results concerning the Hausdorff dimension of these sets starting with those of Baker in 1975 and continuing to recent work of Bishop. In particular, Baranski, Karpinska, and Zdunik proved that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic function with a particular type of domain called a logarithmic tract is greater than one. We discuss generalizing this result to meromorphic maps with a simply connected direct tract and certain restrictions on the singular values of these maps. In order to accomplish this, we develop tools from Wiman-Valiron theory, showing that some tracts contain a dramatically larger disk about maximum modulus points than previously known. Wiman-Valiron discs and the Hausdorff dimension of Julia sets of meromorphic functions | 11 February 2020 |

Tania Benitez Lopez (University of Liverpool) | **Abstract:** Recently Rempe-Gillen gave an almost complete description of the possible topology of the Julia continua of disjoint-type functions combining well-studied concepts from continuum theory and new techniques of transcendental dynamics. In particular, he constructed a function where all Julia continua are pseudo-arcs, however this arises as a special case of a more general construction.
In this presentation, we discuss how to construct a disjoint-type function such that all Julia continua are pseudo-arcs using a different technique which is more explicit, and as a result we obtain better control over the lower order of growth of the function. Julia continua of transcendental entire functions | 4 February 2020 |

Reem Yassawi (The Open University) | **Abstract:** Let (X,σ) be a topological dynamical system, where X is a compact metric space and σ:X → X is a homeomorphism. Its Ellis semigroup is the compactification of the group action generated by σ in the topology of pointwise convergence on the space X^{X}. The Ellis semigroup is, typically, a huge beast, and its computation has been restricted mainly to systems (X,σ) which are metrically equicontinuous; such systems are called tame.
In this talk we give a complete description of the Ellis semigroup for the family of bijective substitution shifts (X,σ). These systems are not tame. (X,σ) admits an equicontinuous factor π: (X,σ)→(Y,δ), and so the Ellis semigroup E(X) is an extension of Y by its subsemigroup E^{fib}(X) of elements which preserve the fibres of π; this includes all idempotents. We give a complete description of E^{fib}(X), expressing it as an uncountable product of the finite group G, defined to be the normal closure of the group generated by the idempotents, with a semigroup Σ. We illustrate with examples all the possibilities that can occur. This is joint work with Johannes Kellendonk. The Ellis semigroup of bijective substitution shifts | 28 January 2020 |

Clodoaldo Ragazzo (University of São Paulo) | **Abstract:** I will present the basic elements of the theory of
tidal deformations. I will show how
different visco-elastic models can be used to describe the rheology of a celestial
body. The results are presented by means of a concrete application: the analysis of the libration of Enceladus due to Saturn. Modelling of tidal forces with application to the librations
of Enceladus. | 28 January 2020 |

Ana Rodrigues (University of Exeter) | **Abstract:** In this talk I will discuss how to compute the entropy following backwards trajectories in a way that at each step every preimage can be chosen with equal probability introducing fair measure and fair entropy (joint with M. Misiurewicz).
I will discuss some advances on the study of fair entropy for non-invertible interval maps under the framework of thermodynamic formalism, showing that the fair measure is usually an equilibrium state (joint with Y. Zhang).
I will then talk about some recent results (joint with S. and Z. Roth) about
transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps. Recent advances on fair measures and fair entropy | 21 January 2020 |

Fritz Colonius (University of Augsburg) | **Abstract:** The use of digital channels for control systems has led, among others, to the problem to determine minimal data rates needed for performing control tasks like stabilization or invariance. This poses new mathematical challenges. The notion of invariance pressure generalizes invariance entropy by adding a potential on the control range and gives further insight into the problem to make a subset of the state space invariant. This talk is based on joint work with Joao Cossich and Alexandre Santana. Invariance pressure for control systems | 14 January 2020 |

Amie Wilkinson (University of Chicago) | **Abstract:** tba tba | 12 December 2019 |

Peter Tino (University of Birmingham) | **Abstract:** Current Machine Learning approaches cannot easily and naturally handle
temporal data in the form of noisy, sparse and irregularly sampled time
series. We suggest a new model-based framework for formulating machine
learning on such data. The framework is based on the idea of Learning
in the Model space, where each data item (time series) is represented as
the posterior distribution over possible models, given the observations.
The framework is general, but in the talk we will show how to formulate
a classifier operating in the space of posteriors over the models.
Besides being "transparent" (the classifier decisions can be understood
through the baseline inferential model), the framework also allows for a
model space analog of feature extraction - detecting the most important
sub-models relevant for the task at hand. Machine Learning in the Space of Dynamical Systems | 3 December 2019 |

Maik Gröger (University of Vienna) | **Abstract:** In this talk we will be interested in investigating the transient
behavior and the occurrence of
dimension drops for maps on the real line which are skew-periodic
Z-extensions of expanding interval maps.
Our main focus lies in the dimensional analysis of the recurrent and
transient sets as well as in determining
the whole dimension spectrum with respect to alpha-escaping sets. For
doing this, we introduce the concept
of fibre-induced pressure which allows us to express the occurrence and
the height of dimension drops in
our setting exclusively in terms of this newly developed pressure.
Moreover, we are able to show that a
dimension drop occurs if and only if we have non-zero drift. If time
permits, we will also present closely
related new insights concerning transient one-dimensional dynamics with
reflective boundary.
This is joint work with Johannes Jaerisch and Marc Kesseböhmer. Dimension drops, escaping sets and thermodynamic formalism for transient
dynamics on the real line | 26 November 2019 |

Mary Rees (University of Liverpool) | **Abstract:** Bela Kerekjarto was a Hungarian topologist active in the 1930's. He is well-known in some quarters, but I first came across him while researching the work of Yael Naim Dowker, who taught at Imperial from the mid 1950's until her retirement in the mid 1980's. Some of her work, in particular with her Ph D student George Lederer, impinged on topological dynamics. Their joint work used a remarkable result of Kerekjarto which in essence shows the existence of nontrivial stable/unstable sets for any compact invariant set, some 25 years before the publication of the now-classical corresponding results for the differentiable category. I shall discuss the implications for the structure of the set of invariant sets of a homeomorphism of a compact metric space. There will be more questions than answers. Invariant sets from Kerekjarto | 19 November 2019 |

Paul Glendinning (University of Manchester) | **Abstract:** Sebastian van Strien wrote a short paper explaining the role (or lack of it) of robust chaos for smooth maps. I will show that with a little reformulation it can be useful for piecewise smooth maps and fill in some missing pieces from the literature. I will also describe some extensions using ideas of geometric continuity of attractors. Much of this is joint work with David Simpson (Massey, Palmerston North, NZ). Robust Chaos Revisited | 12 November 2019 |

Rhiannon
Dougall (University of Bristol) | **Abstract:** A classical example of an Anosov flow would be the geodesic flow associated to a compact hyperbolic manifold M. The periodic orbits are then closed geodesics in M, and this topic has a rich history. In general Anosov flows are not so well behaved, there may be infinitely many periodic orbits in a free homotopy class, in contract to geodesic flows. Nevertheless one has results on the asymptotic number of periodic orbits up to period T. In this talk we will discuss the problem of counting periodic orbits in infinite covering manifolds, where we relate the exponential growth rate of periodic orbits in the cover to properties of the covering group. This is joint work with Richard Sharp. Growth of periodic orbits for Anosov flows in covering manifolds and amenability | 5 November 2019 |

Lasse Rempe-Gillen (University of Liverpool) | **Abstract:** In this talk, we consider the following question. Suppose that we glue a (finite or infinite) collection of equilateral triangles together in such a way that each edge is identified with precisely one other edge, each vertex is identified with only finitely many other vertices. If the resulting surface is connected, it naturally has the structure of a Riemann surface, i.e., a one-dimensional complex manifold. We ask which surfaces can arise in this fashion.
The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface can arise from this construction if and only if it is defined over a number field. These Belyi surfaces and their associated “dessins d’enfants” have found applications across many fields of mathematics, including mathematical physics.
In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces). The talk should be accessible to a general mathematical audience, including postgraduate students. Building surfaces from equilateral triangles | 31 October 2019 |

Simon Baker (University of Birmingham) | **Abstract:** A well known result due to Koksma states that for Lebesgue almost every x>1 the sequence (x^n) is uniformly distributed modulo 1. In this talk I will discuss a recent refinement of this result. Namely that for Lebesgue almost every x>1 the sequence (x^n) has Poisonnian pair correlations. This is based upon some joint work with Christoph Aistleitner. Powers of real numbers | 29 October 2019 |

Disheng Xu (Imperial College London) | **Abstract:** This will be an introductory talk.
The centralizer Z(f) of a diffeomorphism f is the set of diffeomorphisms g that commute with f. In other words, Z(f) is the group of symmetries of f, where "symmetries" is meant the classical sense: coordinate changes leave the dynamics of the system unchanged. For certain algebraic examples, they may have exceptional large symmetry group, i.e. the centralizer contains a non-trivial Lie group. For example, the centralizer of the time-1 map f_0 of the geodesic flow of a negatively curved surface contains R, etc.
A natural question is: how about the symmetry group of a perturbation of f_0? This relates to one of the classical questions in perturbation theory: if a diffeomorphism belongs to a smooth flow, which perturbations also belong to a smooth flow. In this talk we will show some background knowledge and basic ideas of some recent results in this direction, joint works with D. Damjanovic and with D. Damjanovic, A. Wilkinson. Centralizer of smooth dynamical systems with certain hyperbolicity. | 22 October 2019 |

Vasiliki Evdoridou (The Open University) | **Abstract:** Many authors have studied sets associated with the dynamics of transcendental entire functions which have the topological property of being a spider's web. A spider’s web is a connected set which contains `loops’ that surround each other. We consider the analogue of this structure in the punctured plane, and study its connection with the usual spider's web. The escaping set, which plays a key role in the iteration of transcendental entire functions, is defined for transcendental self-maps of the punctured plane to consist of the points whose orbit accumulates at a subset of {0,infinity}. We construct the first example of a transcendental self-map of the punctured plane whose escaping set is a spider’s web and hence it is connected. This is joint work with D. Marti-Pete and D. Sixsmith. Spiders’ webs in the punctured plane | 15 October 2019 |

Natalia Jurga (University of Surrey) | **Abstract:** We study the top Lyapunov exponents of random products of positive matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott, the algorithm is based on the Fredholm theory of determinants of trace-class linear operators. In this article we obtain a simpler expression for the approximations which only require calculation of the eigenvalues of finite matrix products and not the eigenvectors. Moreover, we obtain effective bounds on the error term in terms of two explicit constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the minimal amount of projective contraction of the positive quadrant under the action of the matrices. This is joint work with Ian Morris. Effective estimates on the top Lyapunov exponent of random matrix products. | 8 October 2019 |

Alexandre Rodrigues (University of Porto) | **Abstract:** In this seminar, we explore the chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions.
We extend previous results on the field and we show that, when the cycle is broken, there are parameters for which the first return map to a given cross section exhibits homoclinic tangencies associated to a dissipative saddle periodic point. These tangencies can be slightly modified in order to satisfy the Tatjer conditions for a generalized tangency of codimension two. This configuration may be seen the organizing center, by which one can obtain strange attractors and infinitely many sinks.
Therefore, the existence of a homoclinic cycle associated to a bifocus may be considered as a criterion for four-dimensional flows to be $C^1$-approximated by other flows exhibiting strange attractors. Strange attractors near a homoclinic cycle to a bifocus | 3 October 2019 |

Andreas Bittracher (Freie Universität Berlin) | **Abstract:** The effective longtime dynamics of complex multiscale systems can often be described by just a small number of essential degrees of freedom, also known as reaction coordinates (RCs). In cases where good RCs are known beforehand, the Mori Zwanzig formalism offers an elegant, self-contained scheme to compute the associated effective dynamics. However, the data-driven derivation of good RCs often proves difficult, in part due to a lack of robust mathematical criteria for judging their quality.
Recently, a new mathematical framework for the characterization and computation of RCs has been developed [1]. The derived RCs are optimal in preserving the longest implied timescales and the associated slow sub-processes of the original system. At the heart of the new theory lies the observation that state space points that are "statistically indistinguishable" under the longtime evolution of the system are "geometrically indistinguishable" after embedding into a certain function space. In this embedding space, the points thus lie on a low-dimensional manifold which can be numerically identified by established manifold learning algorithms. In particular, a newly-developed kernelized variant of the diffusion maps algorithm has proven especially well-suited for the problem [2]. The new theory has deep links to Markov state modelling and transition path theory.
This talk will present both the theoretical concepts of the new method, as well as the data-driven algorithms, and demonstrate them on realistic biochemical systems.
[1] "Transition Manifolds of Complex Metastable Systems". Bittracher, A., Koltai, P., Klus, S., Banisch, R., Dellnitz, M., Schütte, C. J Nonlinear Sci (2018) 28: 471. https://doi.org/10.1007/s00332-017-9415-0
[2] "Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds". Bittracher, A., Klus, S., Hamzi, B., Schütte, C. arXiv Preprint (2019). https://arxiv.org/abs/1904.08622 Dimensionality reduction of complex systems: the transition manifold approach | 1 October 2019 |

Francois Berteloot (Universite Paul Sabatier) | **Abstract:** This will be a survey talk about bifurcations within holomorphic
families
of rational functions.
I will explain how the Lyapunov exponent of the maximal entropy measure,
seen as a function of the parameter,
may allow to obtain different kinds of information about the structure
of the bifurcation locus. Lyapunov exponents and bifurcations in holomorphic dynamics | 23 July 2019 |

Michael Benedicks (Uppsala University) | **Abstract:** In the standard Hénon family various coexistence phenomena can occur. In particular there is a positive Lebesgue measure
set of parameters such that finitely many attractive periodic orbits and
a "strange attractor" coexist. We also get a new approach to the
Newhouse phenomenon of infinitely many coexisting attractive periodic
orbits. Also two strange attractors can coexist for maps with
parameters in the classical Hénon family. This is joint work with
Liviana Palmisano. Coexistence phenomena for Hénon maps | 2 July 2019 |

Ghani Zeghib (ENS, Lyon) | **Abstract:** A conic structure consists in giving a cone on each tangent space of a manifold. It gives naturally rise to two kinds of dynamics, one by seeing it as a set valued dynamical system, and a second by considering the action of its automorphism group. We prove a rigidity result for the latter action stating that if it has has a strong dynamics then the cone structure is quadratic. Automorphism groups of conic structures | 1 July 2019 |

M. Martens & L. Palmisano (Stony Brook & Uppsala Univeristy) | **Abstract:** We prove that the Newhouse phenomenon has a codimension 2 nature. Namely, there exist codimension 2 laminations of maps with infinitely many sinks. The leaves of the laminations are smooth and the sinks move simultaneously along the leaves. These Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. As consequence, in the space of polynomial maps, there are examples of:
1. two dimensional Hénon maps with finitely many sinks and one strange attractor,
2. Hénon maps, in any dimension, with infinitely many sinks,
3. quadratic Hénon-like maps with infinitely many sinks and one period doubling attractor,
4. quadratic Hénon-like maps with infinitely many sinks and one strange attractor,
5. two dimensional Hénon maps with finitely many sinks and two period doubling attractors,
6. quadratic Hénon-like maps with finitely many sinks, two period doubling attractors and one strange attractor.
Newhouse Laminations | 27 June 2019 |

Patrice Le Calvez (IMJ-PRG) | **Abstract:** We prove that if S is a smooth compact boundaryless orientable surface, furnished with a smooth area form, then generically in the space of at least once continuously differentiable diffeomorphisms, preserving this form, there exist hyperbolic periodic points and every hyperbolic periodic point has a transverse homoclinic intersection. Homoclinic intersections for area preserving diffeomorphisms of surfaces | 25 June 2019 |

Dirk Blömker (University of Augsburg) | **Abstract:** For a stochastic partial differential equation we approximate the
infinite dimensional stochastic
dynamics by the motion along a finite dimensional slow manifold.
This manifold is deterministic,
but not necessarily invariant for the dynamics of the unperturbed
equation.
Our main results are the derivation of an effective equation (given by
a stochastic ordinary
differential equations) on the slow manifold,
and furthermore the stochastic stability of the manifold
in the sense that with probability almost 1 the solution stay close to
the manifold for very long times.
We present applications to motion of multiple kinks for the stochastic
one-dimensional Cahn-Hilliard equation,
the motion of a single droplet along the boundary of the domain in the
two- or three dimensional mass-conservative Allen-Cahn equation,
and the motion of droplets in the stochastic Cahn-Hilliard equation.
Approximate slow manifolds for SPDEs | 28 May 2019 |

Xiaolong Li (Huazhong University of Science and Technology) | **Abstract:** For a C^1 diffeomorphism f of a 3-dimensional closed manifold, suppose f has a homoclinic tangency associated to a hyperbolic periodic point p (i.e. the stable manifold of p intersects its unstable manifold non-transversally) whose first return map admits non-real eigenvalues. In this talk, I will show a geometric model of mixing Lyapunov exponents inside the homoclinic class of p. More precisely, if p has stable index two and the sum of its largest two Lyapunov exponents is greater than log(1-\delta), then \delta-weak contracting eigenvalues are obtained by an arbitrarily small C^1 perturbation. Using this result, we give sufficient condition for stabilizing a homoclinic tangency within a given range in the C^1 topology. A geometric model of mixing Lyapunov exponents inside homoclinic classes in dimension three. | 21 May 2019 |

Eddie Nijholt (UIUC) | **Abstract:** Network dynamical systems display many unusual properties that are
unheard of for general ODEs. Examples include complicated bifurcation
scenarios, invariant subspaces and highly degenerate spectra. I will
focus on another peculiarity though, namely the observation that for
many linear network maps the eigenvalues are given by simple, linear
expressions of the coefficients. The goal is to define and generalise
such expressions (called network multipliers) and to show that they may
be used to describe the spectrum of the admissible maps of many networks
simultaneously. Moreover, it will turn out that network multipliers are
linearly independent, multiplicative, and that they describe the
spectrum of a network map for all choices of the internal phase space of
a node at once. This is joint work with prof. Lee DeVille from the UIUC. Simple expressions for the eigenvalues of linear network maps | 7 May 2019 |

Selim Ghazouani (Warwick University) | **Abstract:** In this talk I will be interested in generic properties of families of circle homeomorphisms. More precisely, I will try to address the following question: if I pick a circle homeomorphism at "random", what kind of dynamical behaviour should I expect to observe? Will it be minimal or will it have periodic orbits?
I will try to give some context, building upon celebrated work of Arnold, Herman and Yoccoz dealing with the smooth case and, in a second part, attempt at introducing geometric methods that shed some light on the piecewise affine family. Piecewise affine circle homeomorphisms | 19 March 2019 |

Romain Dujardin (Sorbonne University) | **Abstract:** The J=J* conjecture is an important open problem on the dynamics of complex Hénon mappings. The question is whether the Julia set coincides with the closure of the set of saddle periodic points. In the talk I will present several new results towards this conjecture. Some new results on the J=J* problem | 12 March 2019 |

Paul Verschueren (Imperial) | **Abstract:** In a famous paper from 1930, Hardy & Littlewood published a result in Diophantine Approximation, analysing the growth of a series of cosecants. In "The Collected Papers of GH Hardy", Davenport wrote in an introduction: "The proof of this remarkable result is curiously indirect; it involves contour integration and the use of Cesaro means of arbitrarily high order". He included it in a list of one of the top 5 unsolved problems from Hardy's work: "The problem is to give a simpler and more direct proof of these results".
A breakthrough was made in 2009 by Sinai and Ulcigrai who proved a result on the related series of cotangents using the "cut and stack" technique of interval dynamics. Although elementary, the proof was far from simple. We will present another new approach based on circle dynamics, and discuss the challenges remaining. Diophantine Approximation and Dynamical Systems on Cylindrical Phase Spaces with Singularities | 5 March 2019 |

Ale Jan Homburg (University of Amsterdam) | Asymptotics for a class of iterated random cubic operators (motivated by
models for evolution of frequencies of genetic types in populations) | 5 March 2019 |

Julia Slipantschuk (Queen Mary University of London) | **Abstract:** Using analytic properties of Blaschke factors, we construct families of expanding circle maps as well as families of hyperbolic toral diffeomorphisms for which the spectra of the associated transfer operator acting on suitable Hilbert spaces can be computed explicitly. Spectral data for expanding and hyperbolic maps | 26 February 2019 |

José Alves (University of Porto) | **Abstract:** We consider some classes of piecewise expanding maps in
finite dimensional spaces with invariant probability measures, which
are absolutely continuous with respect to Lebesgue measure. We derive
an entropy formula for such measures. Using this entropy formula, we
present sufficient conditions for the continuity of that entropy with
respect to the parameter in some parametrized families of maps. We
apply our results to some families of piecewise expanding maps. Joint
work with Antonio Pumariño. Entropy formula and continuity of entropy for piecewise expanding maps | 19 February 2019 |

Alex Clark (Queen Mary University of London) | **Abstract:** We will provide some background on tiling spaces, reviewing how they are constructed and support a natural translation action. Then we will review some recent results that relate the dynamics and the topology of these spaces. Tilings spaces: their topology and dynamics | 5 February 2019 |

Andrew Clarke (Imperial College) | **Abstract:** Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved in the 70’s that in two dimensions, it is impossible for this angle to tend to zero. Using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, this phenomenon is generic in the real-analytic category. Arnold Diffusion in Multi-Dimensional Convex Billiards | 29 January 2019 |

Jens Marklof (University of Bristol) | **Abstract:** It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have
so far been restricted to almost sure convergence in the space of polynomials of a given degree. Using effective equidistribution results for unipotent flows, we will here provide explicit Diophantine conditions on the coefficients of polynomials of degree 2, under which the convergence of an averaged pair correlation density can be established. The lecture will also provide a survey of related results for other elementary sequences. The gap distribution for fractional parts of simple sequences | 22 January 2019 |

Artem Dudko (IMPAN) | **Abstract:** Let F(z) be the fixed point of the period doubling renormalization (known as the Feigenbaum map). Recently, jointly with Scott Sutherland we showed that the Julia set of F(z) has Hausdorff dimension strictly less than two (without obtaining any more accurate information). In this talk I will recall the latter result and will present new estimates of this Hausdorff dimension from below. The approach used is applicable to other renormalization fixed points.
On Hausdorff dimension of the Feigenbaum Julia set | 15 January 2019 |

Dierk Schleicher (Jacobs University) | **Abstract:** We discuss Newton’s method as a root finder for polynomials of large degrees, and as an interesting dynamical system in its own right. There is recent progress on Newton’s method in at least three directions:
1. We present recent experiments on finding all roots of Newton’s method for a number of large polynomials, for degrees exceeding one billion, on standard laptops with surprising ease and in short time, with observed complexity $O(d log^2 d)$ (joint with Marvin Randig, Simon Schmitt and Robin Stoll – three high school students at the time!).
2. We outline theory about the complexity of Newton’s method as a root finder: unlike various other known methods, Newton as a root finder has both good theory and good implementation results in practice (partly joint work with Magnus Aspenberg, Todor Bilarev, Bela Bollobas, and Malte Lackmann).
3. We discuss Newton’s method as a dynamical system: if $p$ is a polynomial, then the Newton map is a rational map that very naturally “wants to be iterated”. Among all rational maps, Newton’s method has the best understood dynamics, and these dynamical systems can be classified (in the sense of a theory developed by Bill Thurston). As a byproduct, we offer an answer to a question of Steve Smale on existence of attracting cycles of higher period (joint work with Kostiantyn Drach, Russell Lodge and Yauhen Mikulich). Newton’s method as an unexpectedly efficient root finder and as an interesting dynamical system | 11 December 2018 |

Bastien Fernandez (LPSM - CNRS) | **Abstract:** I will present rigorous results on the dynamics of a piecewise affine system of pulse-coupled oscillators with global interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, the analysis essentially boils down to estimating possible asymptotic distributions of clusters (ie. those groups of oscillators that evolve in sync) depending on the initial conditions. I will show that, as the coupling strength increases, the system exhibits a sharp transition between a regime of arbitrary asymptotic distributions to one where every surviving distribution must contain a giant cluster. I will also report on manifestations of this phase transition in the dynamics of uniformly drawn random initial conditions. The most significant feature is that, for large coupling strength, while the maximum number of asymptotic clusters linearly diverges in the thermodynamic limit, for initial conditions with large probability, this number remains bounded. Joint work with A. Blumenthal and L. Tsimring. Transition to Massive Clustering in Populations of Degrade-and-fire Oscillators | 4 December 2018 |

Erwin Luesink (Imperial College) | **Abstract:** In Holm 2015 a means to introduce noise, called stochastic advection by Lie transport, into models for continuum mechanics was proposed. To see whether this noise is qualitatively different from other types of multiplicative noise, it was inserted into the Rayleigh-Benard convection model and the corresponding stochastic Lorenz 63 system was derived. Via the random dynamical systems framework, this L63 system was compared with a L63 system with multiplicative noise in each variable. It is shown that there is a difference between the two systems via Lyapunov exponents. This is done both analytically as well as numerically. To compute Lyapunov exponents numerically, an adapted Cayley method is used. Lyapunov exponents for two Lorenz 63 systems with multiplicative noise | 27 November 2018 |

Daniel Meyer (University of Liverpool) | **Abstract:** A quasisymmetry maps balls in a controlled manner. These maps are
generalizations of conformal maps and may be viewed as a global
versions of quasiconformal maps. Originally, they were introduced
in the context of geometric function theory, but appear now in
geometric group theory and analysis on metric spaces among
others. The quasisymmetric uniformization problem asks when a
given metric space is quasisymmetric to some model space. Here we
consider ``quasitrees''. We show that any such tree is
quasisymmetrically equivalent to a geodesic tree. Under
additional assumptions it is quasisymmetric to the ``continuum
self-similar tree''. This is joint work with Mario Bonk. Uniformization of quasitrees | 13 November 2018 |

Ian Melbourne (University of Warwick) | **Abstract:** The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic). Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.
In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1Anomalous diffusion in deterministic Lorentz gases | 6 November 2018 |

Xavier Buff (University of Toulouse) | **Abstract:** We study the irreducibility (over Q or over C) of various loci defined dynamically. For example,
given an integer d>=2, consider the space of polynomials f_a(z) = a z^d+1. Is the set of parameters
a such that 0 is periodic with period n irreducible over Q ? As another example, consider the space of
cubic polynomials f_{b,c} which have a critical point at 0 with associated critical value 1 and a critical point
c different from 0 with critical value b different from 1. Is the set of parameters (b,c) such that 0 is
periodic with period n irreducible over C ? Irreducibility in holomorphic dynamics | 30 October 2018 |

Mike Todd (St Andrews) | **Abstract:** The `statistics’ of a dynamical system is the collection of statistical limit laws it satisfies. This starts with Birkhoff’s Ergodic Theorem, which is about averages of some observable along orbits: this is a pointwise result, for typical points for a given invariant measure. Then we can look for forms of Central Limit Theorem, Large Deviations and so on: these are about how averages fluctuate, globally, with respect to the invariant measure.
In this talk I’ll show how the form of the `pressure function' for a dynamical system determines its statistical limit laws. This is particularly interesting when the system has slow mixing properties, or, even more extreme, in the null recurrent case (where the relevant invariant measure is infinite). I’ll start by introducing these ideas for simple interval maps with nice Gibbs measures and then indicate how this generalises. This is joint work with Henk Bruin and Dalia Terhesiu. Phase transitions and limit laws | 23 October 2018 |

Luna Lomonaco (University of Sao Paulo) | **Abstract:** In 1994 S. Bullett and C. Penrose introduced a one complex parameter family of holomorphic correspondences, which we denote F_a, and proved that for every real parameter in the connectedness locus such correspondence is a mating between a quadratic polynomial and the modular group. They conjectured that this is the case for every parameter in the connectedness locus.
We show here that matings between the
modular group and rational maps in the parabolic quadratic family Per_1(1) provide a better model: we prove that every member of the family F_a which has the parameter in the connectedness locus is
such a mating. Moreover, we develop a dynamical theory for such a family which parallels the Douady-Hubbard theory of quadratic polynomials. This is a joint work with S. Bullett. Correspondences in dynamics | 16 October 2018 |

Gabriel Fuhrmann (Imperial) | **Abstract:** A classical result by Markley and Paul states that irregular almost automorphic systems over irrational rotations are typically not uniquely ergodic and have positive entropy. By constructing a particular Cantor set, we prove that for each irrational rotation there still are such almost automorphic extensions which are mean-equicontinuous (and hence have zero entropy and are uniquely ergodic). We will shortly review the results by Markley and Paul and then discuss the construction of the Cantor set.
This is a joint work with Eli Glasner, Tobias Jäger and Christian Oertel. Unique ergodicity and zero entropy of irregular symbolic extensions of irrational rotations | 9 October 2018 |

Tuomas Sahlsten (University of Manchester) | **Abstract:** Quasiregular maps are a good candidate for “natural" generalisations of analytic maps in general Riemannian manifolds. We are interested to do “analytic dynamics” for quasiregular maps (e.g. dimension theory of their Julia sets). However, the distortions of QR maps may not stay under control when one iterates so here it helps to assume the QR maps are Uniformly Quasiregular (UQR). It turns out that having “non-trivial” UQR dynamics on the manifold greatly influences the shape (topology) of the manifold.
In this talk we will in particular compute the entropy of uniformly quasiregular maps for UQR maps on manifolds which are not rational cohomology spheres. The methods of the proof include analysis of the transfer operators for UQR maps arising from the local index and the theory of normal currents from Geometric Measure Theory (Federer et al.).
This is based on a joint work with Ilmari Kangasniemi (Helsinki), Yusuke Okuyama (Kyoto) and Pekka Pankka (Helsinki). Uniformly quasiregular dynamics | 2 October 2018 |

Yang Fei (Nanjing University) | **Abstract:** If a holomorphic function can be locally linearized near an irrational indifferent fixed point, then we call the maximal linearization domain containing this fixed point a Siegel disk. Since Siegel proved the existence of the Siegel disks in 1942, much study has been done on the topological and geometric properties of the Siegel disks. In this talk we will give a survey of the results on the Siegel disks, including three conjectures raised by Douady, Sullivan and Herman in 1980s, and some recent developments on these conjectures. A survey of the results on Siegel disks | 31 July 2018 |

Edson de Faria (University of Sao Paulo) | **Abstract:** We discuss a generalization of topological entropy in which the usual exponential growth-rate function is replaced by an arbitrary gauge function. This generalized topological entropy had previously been described by Galatolo in 2003 – up to a choice of notation in the defining formulas – which in turn is essentially the same as that described by Zhao and Pesin in 2015 (that involves a re-parameterization of time). One of the main motivations for studying this new set of invariants comes from the need to distinguish maps with zero (standard) topological entropy. In such cases, if the dynamics is not equicontinuous, then there exists at least one gauge for which the corresponding generalized entropy is positive. After illustrating this simple qualitative criterion, we perform a more quantitative study of the growth of orbits in some low-dimensional examples of zero-entropy maps. Our examples include period-doubling maps in dimension one, and maps of the annulus built from circle homeomorphisms having an exceptional minimal set. This talk is based on joint work with P. Hazard and C. Tresser. Slow growth and entropy-type invariants | 24 July 2018 |

Janosch Rieger (Monash University) | **Abstract:** In this talk, I will discuss spaces of polytopes with fixed outer normals and their use in theoretical and practical shape optimization. These spaces possess a natural system of coordinates, and all admissible coordinates can be characterized by a linear inequality, which is handy both from an analytical as well as from a computational perspective.
The polytope spaces approximate the space of all nonempty convex and compact subsets in Hausdorff distance uniformly on every bounded set, so they behave like classical Galerkin approximations to function spaces. I will show that for simple shape optimization problems, the set of global minimizers of auxiliary problems posed in the polytope spaces converges to the set of global minimizers of the original problem. A Galerkin-type approach to shape optimisation in the space of convex sets | 10 July 2018 |

Amir Jafarian (UCL) | **Abstract:** In this talk, a mesoscopic model of a cortical column, known as a Duffing Neural Mass Model (DNMM), is developed to emulate stochastic mechanisms of initiation and termination of seizures in intracranial electroencephalogram (iEEG) recordings. The DNMM is constructed by applying perturbations to linear models of synaptic transmission in the Jansen and Rit [1] neural mass model. Random input (noise) can cause switches between normal activity and pathological activity similar to seizures in the DNMM. A bifurcation analysis and simulations are presented to provide insights into the behaviour of the model and to motivate questions for discussion. To replicate the pathological dynamics of ion currents, the model is extended to a slow-fast DNMM by considering a slow dynamics model (relative to the membrane potentials and firing rates) for some internal model parameters. The slow-fast DNMM can replicate initiation and termination of seizures that are caused by both random input fluctuations and pathological dynamics. Model comparison and the most likely to capture the underlying dynamics of recorded iEEG is sought through measuring a likelihood function estimated using a continuous-discrete unscented Kalman filter.
Reference:
[1]. B. Jansen and V. Rit. Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biological Cybernetics, 73:357-366, 1995. ISSN 0340-1200. Duffing Neural Mass Models | 26 June 2018 |

Alexey Kazakov (National Research University Higher School of Economics) | **Abstract:** In this talk we present an example of new strange attractor. We show that it belongs to a class of wild pseudohyperbolic spiral attractors. A theory of pseudohyperbolic spiral attractors was proposed in Turaev & Shilnikov paper in 1998, however examples of concrete systems of differential equations with such attractors were not known. Wild-hyperbolic spiral attractor in four-dimensional Lorenz system | 26 June 2018 |

Patricia Soto (Benemerita Universidad Autonoma de Puebla) | **Abstract:** We will define some concepts of complex dynamics and give some families of holomorphic functions. We will give some results related to the Fatou and Julia sets for some real parameters taken from either a plane of parameters or a cut of the space of parameters. Dynamics of some analytic family of functions | 21 June 2018 |

Pablo Rodriguez-Sanchez (Wageningen University) | **Abstract:** This is a story about multidisciplinarity. It starts with a theoretical physicist being hired as a mathematician by a biology department. But, what is the role of a mathematician in such a singular ecosystem? In this talk, we'll learn that the relation between biology and mathematics can be traced back to the XIII century. We'll also learn that the survival of plankton communities is strongly related with chaotic attractors, and how differential geometry had an unexpected role in a science communication problem. Invasive species: a mathematician among biologists | 19 June 2018 |

Olga Pochinka and Viacheslav Grines (Novgorod State University) | **Abstract:** The problem of finding conditions when Morse-Smale cascade can be
embedded into a topological flow goes back to J. Palis (1969). He
stated certain necessary conditions of embedding of Morse-Smale
cascade into a topological flow for dimesion two or higher.
J. Palis then proved the sufficiency of the conditions for embedding of Morse-Smale diffeomorphism into a flow in dimension two. We show that already in dimension three they are not sufficient.
Our report is devoted to finding of sufficient conditions for embedding of Morse-Smale diffeomorphism of the n-sphere, for n at least three, into topological flow. We discuss an interrelation between these conditions and the necessary conditions obtained by J. Palis.
Our results was obtained in collaboration with E. Gurevich and
V.Medvedev [1, 2, 3]
References
[1] V. Grines, E. Gurevich, V. Medvedev, O. Pochinka, On the embedding of Morse-Smale diffeomorphisms on a 3-manifold in a topological flow, Mat. Sb., 2012, Vol.203, No. 12, 81-104; translation in Sb. Math., 2012, Vol. 203, No. 11-12, 1761 - 1784.
[2] V. Grines, E. Gurevich, O. Pochinka, On embedding Morse-Smale diffeo- morphisms on the sphere in topological flows Russian Mathematical Sur- veys(2016), 71 (6):1146
[3] V. Grines, E. Gurevich, O. Pochinka, On embedding of mul- tidimensional Morse-Smale diffeomorphisms in topological flows. https://arxiv.org/abs/1806.03468. 2018 On Palis Problem of Embedding of Morse-Smale Cascades into Flows | 19 June 2018 |

Bastien Fernandez (CNRS) | **Abstract:** To identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for more than two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive.
This talk aims to reconcile these issues and considers a family of N (an arbitrary integer) globally coupled piecewise expanding individual maps, in the weak coupling regime where the dynamics is uniformly expanding. Based on empirical evidences, mathematical proofs and exact numerical test, I will show that a transition in the asymptotic dynamics occurs for every N, as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps | 22 May 2018 |

Zhiqiang Li (Stony Brook University) | **Abstract:** Analogues of the Riemann zeta function were first introduced into geometry by A. Selberg and into dynamics by M. Artin, B. Mazur, S. Smale, and D. Ruelle. Analytic studies of such dynamical zeta functions yield quantitative information on the distribution of closed geodesics and periodic orbits.
We obtain a Prime Orbit Theorem, as an analogue of the Prime Number Theorem, for a class of branched covering maps on the 2-sphere called expanding Thurston maps. More precisely, we show that the number of primitive periodic orbits of such maps, ordered by a weight on each point induced by a non-constant real-valued Holder continuous function (potential) with respect to so-called visual metrics on S^2 satisfying some additional regularity conditions, is asymptotically the same as the well-known logarithmic integral, with an exponential error term. Such a result follows from quantitative study of the holomorphic extension properties of the associated dynamical zeta functions and dynamical Dirichlet series, and spectral properties of some carefully defined Ruelle operators. The geometric properties of the visual metrics also play an essential role here.
In particular, the above result applies to postcritically-finite rational maps with no periodic critical points. Moreover, we prove that the regularity conditions needed here are generic; and for a Lattes map, a continuously differentiable potential satisfies such a condition if and only if it is not cohomologous to a constant. This is a joint work with T. Zheng.
Prime orbit theorems for expanding Thurston maps | 15 May 2018 |

Tomás Lazaro (Universitat Politecnica de Catalunya) | **Abstract:** Regarding the origins of life and the first primitive replicating systems,
M. Eigen (Nobel Prize in Chemistry, 1967) asserts: â€œThere is neither possible
mechanisms of error correction without large information contents nor possible
large information contents without error correction mechanismâ€. This chicken-and-egg problem (known as Eigenâ€™s Paradox ) states that there exists an error threshold
limiting the length of such molecules under high mutation rates.
So, how evolution could overcome this paradox? How could the first replicating systems increase their information content towards more complex entities?
One of the existent theories (conceived by Manfred Eigen and Peter Schuster) states that
the so-called hypercycles could play a crucial rÃ´le to overcome this error threshold. Hypercycles, i.e. nonlinear catalytic networks, allow an all-species coexistence and could support an information content larger than the one found for a quasispecies-based model.
It is known that hypercycles are sensitive to the so-called parasites and short-circuits. While the impact of parasites has been widely investigated for well-mixed and spatial hypercycles, the effect of short-circuits in hypercycles remains poorly understood.
In this talk we will present, briefly, a dynamical description of two small asymmetric hypercycles with short-circuits.
This is a joint work with E. Fontich, T. Guillamon and J. Sardanyes.
Prebiotic evolution: small hypercycles with shortcircuits | 8 May 2018 |

Ke Wu (ETH Zurich) | **Abstract:** We empirically verify that the market capitalisations of coins and tokens in the cryptocurrency universe follow power law distributions with significantly different values, with the tail exponent falling between 0.5 and 0.7 for coins, and between 1.0 and 1.3 for tokens. With a simple birth-proportional growth-death model previously introduced to describe firms, cities, webpages, etc., we validate the proportional growth (Gibrat's law) of the coins and tokens, and find remarkable agreement between the theoretical and empirical tail exponent of the market cap distributions for coins and tokens respectively. Our results clearly characterizes coins as being "entrenched incumbents" and tokens as an "explosive immature ecosystem", largely due to massive and exuberant ICO activity in the token space. The theory predicts that the exponent for tokens should converge to Zipf's law in the future, reflecting a more reasonable rate of new entrants associated with genuine technological innovations. Apart from this, we develop a strong diagnostic for bubbles and crashes in bitcoin, by analyzing the coincidence (and its absence) of fundamental and technical indicators. Using a generalized Metcalfe’s law based on network properties, a fundamental value is quantified and shown to be heavily exceeded. In these bubbles, we detect a universal super-exponential unsustainable growth, modeled by the Log-Periodic Power Law Singularity (LPPLS) model, which parsimoniously captures diverse positive feedback phenomena. Quantification of the crypto-currency ecosystem: Generalised Zipf law of capitalisations and the hierarchy of bitcoin bubbles | 27 March 2018 |

Georg Ostrovski (DeepMind) | **Abstract:** In recent years, the use of deep neural networks in Reinforcement Learning has allowed significant empirical progress, enabling generic learning algorithms with little domain-specific prior knowledge to solve a wide variety of previously challenging tasks. Examples are reinforcement learning agents that learn to play video games exceeding human-level performance, or beat the world's strongest players at board games such as Go or Chess. Despite these practical successes, the problem of effective exploration in high-dimensional domains, recognized as one of the key ingredients for more competent and generally applicable AI, remains a great challenge and is an active area of empirical research.
In this talk I will introduce basic ideas from Deep Learning and its use in Reinforcement Learning and show some of their applications. I will then zoom in on the exploration problem, and present some of the recent algorithmic approaches to create 'curious' reinforcement learning agents. Exploration in Deep Reinforcement Learning | 22 March 2018 |

Jennifer Creaser (University of Exeter) | **Abstract:**
It is well known that the addition of noise in a multistable system can induce random transitions between stable states. Analysis of the transient dynamics responsible for these transitions is crucial to understanding a diverse range of brain functions and neurological disorders such as epilepsy. We consider directed networks in which each node in the network has two stable states, one of which is only marginally stable. We assume that all nodes start in the marginally stable state and once a node has escaped we assume that the transition times back are astronomically large by comparison. We use first-passage-time theory and the well-known Kramers' escape time to characterize transition rates between the attractors. Using numerical and theoretical techniques we explore how sequential escape times of the network are effected by changes in node dynamics, network structure, and coupling strength. Sequential escapes for network dynamics | 20 March 2018 |

Ale Jan Homburg (VU University Amsterdam) | **Abstract:** We consider a class of skew product maps of interval
diffeomorphisms over the doubling map. The interval maps fix the end
points of the interval.
It is assumed that the system has zero fiber Lyapunov exponent at one
endpoint and zero or positive fiber Lyapunov exponent at the other endpoint.
We discuss the appearance of on-off intermittency.
A main ingredient is the equivalent description in terms of
chaotic walks: random walks driven by the doubling map. On-off intermittency and chaotic walks | 20 February 2018 |

Ben Mestel (The Open University) | **Abstract:** Starting from familiar number expansions (decimal, binary etc), we develop the dynamic expansions that form the basis of general quasiperiodic renormalisation, a review of which forms the second part of the talk. Quasiperiodic renormalisation for general rotation number | 13 February 2018 |

John Roberts (University of New South Wales) | **Abstract:** In the past decade, the research area of arithmetic dynamics has grown in prominence. This area considers iterated maps as dynamical systems, acting on the integers, the rationals or on finite fields (meaning there is a finite phase space in the last case). Tools used to investigate arithmetic dynamics include combinatorics, arithmetic geometry, number theory, graph theory as well as numerical experimentation. There are important applications of arithmetic dynamical systems in cryptography. I will survey some of our investigations in arithmetic dynamics which have been motivated by the order and chaos divide in Hamiltonian dynamics. Order and randomness in dynamics over finite fields | 13 February 2018 |

Shaun Bullett (Queen Mary University of London) | **Abstract:** In an article in Inventiones in 1994, Christopher Penrose and I introduced a one-parameter family of 2-to-2 holomorphic correspondences which for certain restricted parameter values we proved to be matings
between the modular group PSL(2,Z) and quadratic polynomials. Experimental evidence supported a much more general picture, but we did not have the technical tools to prove our conjectures.
Twenty years later, Luna Lomonaco introduced a theory of "parabolic-like mappings" which turned out to be just what was needed. I will outline the results Lomonaco and I have jointly obtained, and the techniques we use to prove them: these range from symbolic dynamics to hyperbolic and quasiconformal geometry, and exploit parallels between the behaviour of rational maps and Kleinian groups ("Sullivan's dictionary"). In a final section, I will focus on a new Pommerenke-Levin-Yoccoz inequality which is key to proving that the "Mandelbrot set for matings" is homeomorphic to the classical Mandelbrot set for quadratic polynomials. Matings between quadratic maps and the modular group | 30 January 2018 |

Sanjeeva Balasuriya (University of Adelaide) | **Abstract:** Uncertainties in velocity data are often ignored when computing Lagrangian particle trajectories of fluids. Modelling these as noise in the velocity field leads to a random deviation from each trajectory. This deviation is examined within the context of small (multiplicative) stochasticity applying to a two-dimensional unsteady flow operating over a finite-time. It is proven that the deviation's expectation is zero, and that its variance is bounded by a quantity defined to be the stochastic sensitivity. The stochastic sensitivity field provides a measure of uncertainty for trajectories beginning at each location. An easily computable expression for the stochastic sensitivity is derived. Monte Carlo simulations are used on a model flow to quantitatively validate the usage of the stochastic sensitivity, and moreover demonstrate the significant potential of stochastic sensitivity as a new tool for identifying coherent structures in flows even when the velocity is only available as data. Stochastic sensitivity: a computable measure for uncertainty of deterministic trajectories | 23 January 2018 |

Andrey Shilnikov (Georgia State University) | **Abstract:** We discuss several models, phenomenological and biologically plausible, that undergo a torus bifurcation and torus breakdown
at the transition between neural activity types.
Torus bifurcation in neuroscience models | 23 January 2018 |

Victor Beresnevich (University of York) | **Abstract:** Techniques from the homogeneous dynamics have been a powerful resource for solving several long standing problems in the theory of Diophantine approximation. In this talk I will explain the meaning of the non-divergence of lattices under the action by diagonal matrices, its relationship to Diophantine approximation, the quantitative non-divergence result of Kleinbock and Margulis from 1998 and its role in a recent work on badly approximable points on nondegenerate submanifolds of a finite dimensional Euclidian space. Homogeneous dynamics, Quantitative non-divergence and Diophantine approximation on manifolds | 16 January 2018 |

Jacek Cyranka (UC San Diego) | **Abstract:** We present a computer-assisted proof of heteroclinic connections in the one-dimensional
Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential
equation subject to homogeneous Neumann boundary conditions, which contains as a special case the
celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely
understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer
long-term dynamical behavior, which includes a high level of multistability. We establish the existence of certain heteroclinic connections between the homogeneous equilibrium state and local and global energy minimizers, which constitutes the first in our knowledge constructive proof of the existence of heteroclinic connections for a parabolic PDE. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic at various nontrivial parameter values. Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model | 9 January 2018 |

Camille Poignard (ICMC University of Sao Paulo) | **Abstract:** I will present some investigations made on the study of the synchronization of networks with respect to their structures. The results concern three different settings.
In the first part, we study how the synchronizability of a diffusive network increases (or decreases) when we add some links in its underlying graph. This is of interest in the context of power grids where people want to prevent from having blackouts. We show some classification results obtained (with Tiago Pereira and Philipp Pade) with respect to the effects of these links.
In a second part (in a slightly different context) I will present an example of network exhibiting a chaotic behavior when we add links to its two very stable components.
Lastly, if there is still time for it, I will present a synchronization result (not in continuous but in discrete time) obtained for massively coupled networks having an infinite number of nodes.
References:
C.P., Tiago Pereira, Jan Philipp Pade. Laplacian matrices of weighted graphs: Structural genericity properties. SIAM Journal on Applied Mathematics. 2017. (available on arxiv)
Jan Philipp Pade, C.P., Tiago Pereira. The effects of structural perturbations on the Synchronizability of diffusive networks. arxiv. 2017.
C.P. Inducing chaos in a gene regulatory network by coupling an oscillating dynamics with a hysteresis type one. Journal of Mathematical Biology. 2013.
C.P. Discrete synchronization of massively coupled networks by hierarchical couplings.
Physica D. 2015.
Synchronization and structures of Networks: some results | 8 January 2018 |

Edward Hooton (University of Texas at Dallas) | **Abstract:** Pyragas time-delayed feedback control has proven itself as an effective tool to noninvasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings. In this talk, we consider O4-symmetric systems of van der Pol and optical oscillators coupled in a cube-like configuration. These systems undergo equivariant Hopf bifurcations giving rise to multiple branches of unstable periodic solutions. We introduce a delayed control term, which ensures stabilization of a selected branch. Group theoretic restrictions which help to shape our choice of control are discussed. Furthermore, we explicitly describe the domains in a two-dimensional parameter space for which the periodic solutions of the delayed system are stable. Noninvasive Stabilization of Periodic Orbits in O4-Symmetrically Coupled Systems Near a Hopf Bifurcation point | 12 December 2017 |

David Sixsmith (University of Liverpool) | **Abstract:** Suppose that f is a transcendental entire function, and that U and V are disjoint simply-connected domains. In 1970 it was (implicitly) claimed that at least one of these domains has disconnected pre-image under f. It was recently observed that there is a flaw in the proof of this claim. We discuss this flaw, and how to construct examples which show that the claim is in fact false; indeed, there can be infinitely many disjoint simply-connected domains with connected pre-image under f. We also consider additional hypotheses sufficient to make the claim true. On domains with connected pre-image | 5 December 2017 |

Corinna Ulcigrai (University of Bristol) | **Abstract:** We will present an instance of the central limit theorem in entropy zero dynamics obtained as a temporal limit theorem. We consider deterministic random walks on the real line R driven by a rotations (or in other words, a skew product over an irrational rotation) and prove a temporal CLT for badly approximable rotation numbers and piecewise cocycle with jumps at certain irrational values. This generalizes previous results by J.Beck and by D. Dolgopyat and O. Sarig. The proof uses renormalization in the form of the continued fraction algorithm and Ostrowsky renormalization. The talk is based on joint work with Michael Bromberg. A CLT for cocycles over rotations | 21 November 2017 |

Thomas Wanner (George Mason University) | **Abstract:** The diblock copolymer equation models phase separation processes which involve
long-range interactions, and therefore promote the formation of fine structure.
While the model arises through a regular perturbation from the classical
Cahn-Hilliard model for phase separation in binary alloys, its dynamics is
considerably richer, and exhibits for example a high level of multistability.
As a dissipative model, its long-term dynamics can in principle be completely
described by the dynamics on its global attractor, which is comprised of
equilibrium solutions and connecting orbits between them. Unfortunately, however,
classical mathematical methods have so far failed at uncovering this attractor
structure. In this talk, we provide an overview of how rigorous computational
techniques can be used to obtain computer-assisted proofs for the existence of
equilibrium solutions, curves of secondary bifurcation points, as well as
heteroclinic connections. In the course of this, we uncover the formation
of energy minimizers with fine structure through a homotopy from the classical
Cahn-Hilliard bifurcation diagram, and it will be shown that typical solutions
originating close to the homogeneous state are trapped by local minimizers of
the energy, and do not in fact reach the global minimizers. Bifurcation diagram verification for the diblock copolymer model | 21 November 2017 |

Daniil Yurchenko (Heriot-Watt University) | **Abstract:** Energy harvesting (EH) remains an attractive field of interests due to the needs
for powering low-energy- consumption industrial sensor networks, wearable
sensors or wireless sensors that required power but placed in hard-to- reach
locations, like tires. Besides, enormous efforts undertaken by the scientists all
over the world have not led to a reasonable device design that can be directly
adapted by an industry, although these efforts have resulted in a high number
of publications and significant advances achieved in understanding various
types of mechanical energy conversion. Among many possible ways of
converting mechanical energy of vibrations one may consider a vibro-impact
(VI) interaction, which potentially has a number of benefits like high kinetic
energy at impacts, possible low frequency impact motion and others. Some
ideas have been proposed and investigated on how to incorporate the VI
dynamics into EH process. The focus of this talk is on the application of the VI
dynamics in a combination with dielectric elastomers used as a material for
membranes subjected to the impacts. The design details, the EH principle, VI
dynamics of the device and its applications will be discussed. Vibro-impact processes for energy harvesting | 14 November 2017 |

Samuel Roth (Silesian University) | **Abstract:** How can we interpret the infimum of Lipschitz constants in a conjugacy class of interval maps? For positive entropy maps, the exponential of the topological entropy gives a well-known lower bound. We show that for piecewise monotone maps, these two quantities are equal, but for countably piecewise monotone maps, the inequality can be strict. Moreover, in the transitive and Markov case, we characterize the infimum of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map. Dynamically, this number represents the exponential growth rate of the number of iterated preimages of nearly any point. On Lipschitz Constants and Entropy | 13 November 2017 |

Zuzana Roth (Silesian University) | **Abstract:** (joint work with Samuel Roth)
Akin and Kolyada in 2003 [1] conjectured that every minimal system with a
weak mixing factor, is Li-Yorke sensitive. Our interest in this problem comes from
a survey paper by Li and Ye [2] which came out last year.
This talk will deal with a proof of the conjecture for minimal 2-point extensions
of weak mixing systems, which became an inspiration for a more complex solution
by Mlichova [3]. The talk will be closed with results which showed up in the last
year.
References
[1] Ethan Akin and Sergii Kolyada. Li-Yorke sensitivity. Nonlinearity, 16(4):1421, 2003.
[2] Jian Li and Xiangdong Ye. Recent development of chaos theory in topological dy-
namics. Acta Mathematica Sinica, English Series, 32(1):83–114, 2016.
[3] Michaela Mlichová. Li-Yorke sensitive and weak mixing dynamical systems. Journal
of Difference Equations and Applications, 0(0):1–8, 0.
[4] Song Shao and Xiangdong Ye. A non-PI minimal system is Li-Yorke sensitive. Proc.
Am. Math. Soc., published electronically, Sept. 2017, doi.org/10.1090/proc/13779 Li-Yorke sensitivity and a conjecture of Akin and Kolyada | 13 November 2017 |

Pietro-Luciano Buono (University of Ontario Institute of Technology) | **Abstract:** I will be presenting a family of hyperbolic non-local models for animal
aggregation developed recently (Eftimie, Fetecau). I will be showing that these
models possess nontrivial symmetry groups which one can use to classify the
possible states of the system according to their isotropy subgroups. In particular,
I will discuss the patterns appearing near a codimension two O(2) symmetric
steady-state/Hopf bifurcation. Symmetry-breaking bifurcations in animal aggregation models | 10 November 2017 |

Yusuke Okuyama (Kyoto Institute of Technology) | **Abstract:** For each integer d>1, the "dynamical moduli space" M_d of rational functions
of degree =d is a 2d-2(>1) dimensional complex analytic orbifold generalizing
the parameter space (indeed the complex affine line) of the family of
monic centered quadratic polynomials z^2+c. In this moduli space M_d,
there are not only plenty of "dynamically meaningful" complex hypersurfaces,
which are not necessary irreducible, but also the "bifurcation" measure,
which generalizes the harmonic measure on the boundary of the Mandelbrot set
associated to z^2+c. In this talk, we will discuss on the interaction
between the intersections of dynamical hypersurfaces and
the bifurcation measure in M_d, from the viewpoint of quantitative
equidistribution and counting. This talk is based on a joint work
with Thomas Gauthier and Gabriel Vigny (Amiens, France). Intersection and bifurcation in the dynamical moduli space
of rational functions | 2 November 2017 |

Peyman Eslami (University of Warwick) | **Abstract:**
For a large class of piecewise expanding maps of metric spaces we show the equidistribution of standard pairs at an exponential rate. As a
corollary such systems have a unique absolutely continuous invariant measure with respect to which the system is mixing. We allow for unbounded, non-compact spaces, countably many branches and do not assume big images or the existence of a Markov structure. We show how to control the complexity growth of the dynamical partition of the map. Such control is necessary and crucial for systems that are not one-dimensional. Our method gives explicit estimates on the exponential rate of equidistribution. If there is time, I will also comment on the construction of Banach spaces (made out of standard pairs) on which the transfer operator admits is quasi-compact. If one is not concerned with explicit bounds on the constants involved in decay of correlations, this functional analytic point of view leads to establishing further statistical properties of the system in a standard manner.
Exponential equidistribution of standard pairs for piecewise expanding maps of metric spaces | 31 October 2017 |

Scott Sutherland (Stony Brook University) | **Abstract:** The Feigenbaum polynomial $p_f(z)=z^2 - 1.40155\cdots$ was
discovered independently by Coullet-Tresser and Feigenbaum in the 1970s
as the limit of the period-doubling cascade in the quadratic family, and
giving rise to the universality conjecture and the development of
renormalization theory in real and holomorphic dynamics. Many striking
results regarding this and related polynomials have been proven by
Lanford, Sullivan, McMullen, and Lyubich, among many others.
Recently, Avila and Lyubich constructed examples of quadratic
Feigenbaum polynomials (a more general class in which $p_f$ a member)
which contains quadratic polynomials with Julia sets of positive
measure, as well as those with Hausdorff dimension less than two.
However, the question regarding the area of the Julia set of $p_f$
itself remained open.
Using computer-assisted methods we show that the Julia set of the
Feigenbaum polynomial has Hasdorff dimension less than two (and hence
zero Lebesgue measure). This is based on joint work with Artem Dudko. On the measure of the Feigenbaum Julia set | 26 October 2017 |

Bernd Krauskopf (University of Auckland) | **Abstract:**
The Lorenz system, which has a phase space of dimension three, is well know as a prototypical example of a continuous-time system with a chaotic attractor. Are there even more complicated types of chaotic dynamics in higher dimensions? The answer is yes, as was shown recently by the construction of a five-dimensional Lorenz-type system; one also speaks of wild chaos. I will briefly review classical Lorenz chaos and then present how wild chaos arises in the higher-dimensional context. The latter is associated with several types of bifurcations of a noninvertible map of the punctured plane, which contains the complex quadratic family as a special case.
This is joint work with Hinke Osinga and Stephanie Hittmeyer. Lorenz chaos: now even wilder
| 12 October 2017 |

Genadi Levin (The Hebrew University of Jerusalem) | **Abstract:** We present a general approach to show that critical relations of locally holomorphic maps on the complex plane unfold transversally in a "positively
oriented" way. In one-parameter interval families this property implies the monotonicity of kneading sequence and topological entropy. We mainly illustrate this approach on a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps,
maps with discontinuities but also for families of maps with complex analytic
extensions such as certain polynomial-like maps. Joint work with Weixiao Shen and Sebastian van Strien. Monotonicity of entropy and positively oriented transversality for families of interval maps | 3 October 2017 |

Daniel Thompson (Ohio State University) | **Abstract:** We establish results on uniqueness of equilibrium states for geodesic flows on rank one manifolds. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The geodesic flow on a rank one manifold is a primary example of a non-uniformly hyperbolic flow and I'll explain why it satisfies our hypotheses. The main points are to show that uniqueness holds when the pressure on the singular set is less than the pressure on the whole space; and then to verify this pressure gap condition for natural classes of potentials. This is joint work with Keith Burns (Northwestern), Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young). Uniqueness of equilibrium states for geodesic flows in manifolds of nonpositive curvature | 4 July 2017 |

Nikolaos Karaliolios (Imperial College London) | **Abstract:** We discuss the solvability of the linear cohomological equation over a smooth volume preserving diffomorphism of a smooth compact manifold $(M,\mu )$. The central conjecture in the subject was posed by M. Herman, and states that the only diffeomorphisms for which the equation is solvable for all observables in $C^{\infty}_{\mu } (M, \mathbb{R})$ are Diophantine rotations in tori. Recent attempts to construct counterexamples to the conjecture were based on the Anosov-Katok construction, and went only as far as constructing Distributionally Uniquely Ergodic (DUE) diffomorphisms in manifolds that are not tori. For DUE diffeomorphisms, the cohomological equation is solvable for observables in a dense subspace of $C^{\infty}_{\mu } (M, \mathbb{R})$. We will discuss the problem and propose an argument showing that the Anosov-Katok method cannot provide counterexamples to the conjecture. Cohomological stability and the Anosov-Katok construction | 4 July 2017 |

Marco Martens (Stony Brook University) | Invariant manifolds for renormalisation | 27 June 2017 |

Liviana Palmisano (University of Bristol) | The rigidity conjecture | 27 June 2017 |

Vasileios Basios (Université Libre de Bruxelles) | Linear and nonlinear Arabesgues: Negative 2-element Circuits and Chaos | 21 June 2017 |

Rene Medrano (Universidade Federal de São Paulo) | **Abstract:** TBA Shrimps, cockroaches and some other strange structures in chaotic systems | 20 June 2017 |

Maciej Capinski (AGH University of Science and Technology) | **Abstract:** TBA A fast diffusion mechanism with application to the Neptune-Triton elliptic restricted three body problem | 20 June 2017 |

De-Qi Zhang (National University of Singapore) | **Abstract:** We consider automorphism g of positive entropy on a compact Kahler manifold X.
The existence of such g imposes
strong constraints on the geometrical structure of X. We also show that a projective manifold X has a maximal number of commutative automorphisms
of positive entropy only when X is a complex torus or its quotient. The minimal model program of algebraic geometry will be employed. Automorphisms of positive entropy on
compact Kahler manifolds | 13 June 2017 |

Artur Oscar Lopes (Universidade Federal do Rio Grande do Sul) | New results on Thermodynamic Formalism: entropy, pressure and the involution kernel | 11 April 2017 |

Dmitry Treschev (Steklov Institute, Moscow) | Arnold diffusion in a priori unstable case | 23 March 2017 |

Ilya Goldsheid (Queen Mary University of London) | **Abstract:** The asymptotic behaviour of products of independent identically distributed $m\times m$ random matrices is now relatively well understood (if $m$ is fixed!).
A long standing natural problem is: what part of the corresponding theory can be extended to the case of products of non-identically distributed matrices and, more generally, transformations?
Perturbation theory is a very natural example of a situation where such a question arises.
In my talk, I'll try to answer this question. Products of random transformations and Lyapunov exponents | 21 March 2017 |

Manuela Aguiar (University of Porto) | **Abstract:** A coupled cell system is a dynamical system distributed over the nodes (cells) of a network. Each cell is an individual dynamical system (which we are going to assume continuous) and the coupling structure of the network indicates the interactions between those cell dynamics. One of the key aspects in the theory of coupled cell networks concerns the existence of synchrony subspaces - subspaces defined in terms of equalities between cell coordinates which are flow-invariant. Synchrony subspaces (flow-invariant subspaces) can have a major impact on both global and local dynamics and are important from the point of view of the study of that dynamics. Surprisingly, synchrony subspaces are independent of the specific individual dynamics at the nodes and are determined only by the network structure. That is, all the coupled cell systems admissible by a given network structure share the same structure of patterns of synchrony. In my talk, I will review recent concepts and results concerning and related with synchrony subspaces in coupled cell networks.
The talk includes joint work with Peter Ashwin (University of Exeter), Ana Dias (Univer- sity of Porto), Flora Ferreira (University of Porto), Mike Field (Rice University, Imperial College), Marty Golubitsky (The Ohio State University), Maria Leite (University of South Florida) and Haibo Ruan (Universität Hamburg). About Patterns of Synchrony in Coupled Cell Networks | 21 March 2017 |

Tomoo Yokoyama (Kyoto University of Education) | **Abstract:** In this talk, we present a necessary and sufficient
condition for the existence of dense orbits of continuous flows on
compact connected surfaces, which is a generalization of a necessary
and sufficient condition on area-preserving flows obtained by H.
Marzougui and G. Soler López. We also consider what class of flows on
compact surfaces can be characterized by finite labeled graphs. In
particular, a class of surface flows, up to topological conjugacy,
which contains both the set of Morse Smale flows and the set of
area-preserving flows with finite singular points is classified by
finite labeled graphs. Finally, we discuss applications to fluid
dynamics. Topological transitivity and representability of surfaces flows | 15 March 2017 |

Oliver Jenkinson (Queen Mary University of London) | **Abstract:** The Lagarias-Wang finiteness conjecture asserted that the joint spectral radius of a set of matrices is always realised by a finite matrix product. Although this conjecture has been disproved, counterexamples are not easy to find.
In this talk I will describe a new approach to generating finiteness counterexamples, making links with ergodic optimization and Sturmian measures.
This is joint work with Mark Pollicott. Joint spectral radius, Sturmian measures, and the finiteness conjecture | 14 March 2017 |

Ursula Hamenstädt (Rheinische Friedrich-Wilhelms-Universität Bonn ) | **Abstract:** We begin with looking at a cocycle over the geodesic flow on a negatively curved surface with values in Rn, where the flow is equipped with an invariant measure obtained from a harmonic measure of a random walk. We summarize work of Guivarch Raugy and Benoist Quint which gives a simple criterion for simplicity of the Lyapunov spectrum. We then explain why this covers a very general
class of invariant measures and then show how this viewpoint largely generalizes to many other flows which are not necessariy Anosov flows on a compact manifold. Simplicity of the Lyapunov spectrum for cocycles of flows | 14 March 2017 |

Maik Gröger (University of Jena) | **Abstract:** Studying low-complexity notions and possible concepts of long-rage
order are two closely linked endeavours. With this in mind we
investigate the relations of two complexity notions in the zero
entropy regime: mean equicontinuity and amorphic complexity. As it
turns out, there is a close relationship in the minimal setting and
for mean equicontinuous subshifts we show that amorphic complexity
corresponds to the box dimension of the maximal equicontinuous factor.
Further, for certain Toeplitz subshifts we show how to calculate
amorphic complexity using the theory of iterated function systems. We
will also elaborate on possible extensions to more general group
actions and applications to the theory of quasicrystals. This is work
in progress with Gabriel Fuhrmann, Tobias Jäger and Dominik Kwietniak. Amorphic complexity and long-range order | 8 March 2017 |

Ian Melbourne (University of Warwick) | **Abstract:** Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for the classical Lorenz attractor. Many of the proofs use the smoothness of the stable foliation for the flow, or at least smoothness of the stable foliation for a Poincare map.
In this talk, I will survey the recent results in this area for the classical Lorenz attractor (where the stable foliations are known to be at least $C^{1.278}$), and more generally for singular hyperbolic attractors when the stable foliations are assumed to be $C^{1+\epsilon}$. Then I will describe some new results, joint with Vitor Araujo, where we show that many statistical properties persist without these smoothness assumptions. Such properties include existence of SRB measures, central limit theorems and invariance principles, mixing, and (typically) superpolynomial decay of correlations. However exponential decay of correlations remains an open question in this generality. Mixing etc for Lorenz attractors with nonsmooth stable foliations | 28 February 2017 |

Alexey Kazakov (Nizhny Novgorod University) | Variety of strange attractors in nonholonomic mechanics | 28 February 2017 |

Bassam Fayad (Institut de Mathématiques de Jussieu-Paris Rive Gauche) | **Abstract:** TBA Dynamics and the distribution of $n\alpha$ on the torus | 22 February 2017 |

Jordi-Lluis Figueras (Uppsala University) | **Abstract:** In this talk we will present a numerical algorithm for the
computation of (hyperbolic) periodic orbits of the 1-D
Kuramoto-Sivashinsky equation
u_t+v*u_{xxxx}+u_{xx}+u*u_x = 0,
with v>0.
This numerical algorithm consists on applying a suitable quasi-Newton scheme.
In order to do this,
we need to rewrite the invariance equation that must satisfy
a periodic orbit in a form that
its linearization around an approximate solution
is a bounded operator. We will also show how this methodology
can be used to compute a-posteriori estimates of the errors of the
solutions computed, leading to the rigorous verification of the existence of the periodic orbit.
If time permits, we will finish showing some numerical outputs of the algorithms presented along
the talk.
This is a joint work with Rafael de la Llave, School of Mathematics, Georgia Institute of Technology.
Numerical algorithms and a-posteriori verification of periodic
orbits of the Kuramoto-Sivashinsky equation | 21 February 2017 |

Matteo Ruggiero (Paris 7) | **Abstract:** We consider the local dynamical system induced by a non-invertible selfmap f of C^2 fixing the origin.
Given a modification (composition of blow-ups) over the origin, the lift of f on the modified space X defines a meromorphic map F.
We say that F is algebraically stable if for every compact curve E in X, its orbit through F does not intersect the indeterminacy set of F.
We show that, starting from any modification, we can also blow-up some more and obtain another modification for which the lift F is algebraically stable.
The proof relies on the study of the action f_* induced by f on a suitable space of valuations V.
In particular we construct a distance on V for which f_* is non-expanding. This allows us to deduce fixed point theorems for f_*.
If time allows, I will comment on the recent developments about local dynamics on normal surface singularities.
Joint work with William Gignac. Local dynamics of non-invertible selfmaps on complex surfaces | 14 February 2017 |

Stefan Ruschel (TU Berlin) | **Abstract:** Infectious diseases are among the most prominent threats to mankind.
When an outbreak is not foreseen, preventive healthcare cannot be
provided. In this case, the unanimous means of control is isolation of
infected individuals, as implemented in the 2014 Ebola outbreak in
West Africa. We investigate how isolation of identified individuals
impacts the spread of an otherwise endemic disease. We model this
effect in a homogenous population with mass-action-type contacts. We
obtain a mean field description obeying a system of Delay Differential
Equations, assuming the identification and isolation time are the same
for all individuals. Our analysis reveals that isolation before a
critical identification time prevents the outbreak. When isolation is
implement after this critical time the disease is endemic, but its
severity depends on the time spend in isolation. At first, increasing
the isolation time reduces the number of infected individuals.
However, long isolation causes the disease to reappear periodically
with severe outbreaks. SIQ epidemiological model (Impact of isolation on endemic diseases)
| 7 February 2017 |

Matthieu Astorg (Université d'Orleans) | **Abstract:** Finite type maps are a class of analytic maps on complex 1-manifolds introduced by Epstein, that notably include rational maps and entire functions with a finite singular set. Each of those maps possess a natural finite-dimensional moduli space, and one can define a dynamical Teichmüller space parametrizing their quasiconformal conjugacy class. Using the fact that this Teichmüller space immerses into the moduli space, we will generalize rigidity results of Avila, Dominguez, Makienko and Sienra under an assumption of expansion along the critical orbits. Summability condition and rigidity for finite type maps | 31 January 2017 |

Nikolaos Karaliolios (Imperial College London) | **Abstract:** We revisit M. Herman's KAM theorem on perturbations of Diophantine translations in tori of arbitrary dimension. We give a partially optimal condition for the perturbation to be smoothly conjugated to the exact model and relate this rigidity result with a Denjoy-like construction of McSwiggen. This is part of an on-going project joint with A. Kocsard, where we try to blow up the rotation set by perturbing a pseudo-rotation, and with S. van Strien, where we explore the possibility of generalizing Denjoy's theory to tori of dimension higher than one. Local rigidity for Diophantine translations on tori of arbitrary dimension | 24 January 2017 |

Disheng Xu (Institut de Mathématiques de Jussieu-Paris Rive Gauche) | **Abstract:** We study smooth group actions and random walks on surface under a mild assumption called "weakly expanding". In particular, we prove that some statistical properties (large deviation, equidistribution, etc.) for non-abelian semigroup of linear action on the torus persist under C^1 conservative perturbation of the generators. In addition we give a sufficient condition and an example for robust minimality of the action of semigroup generated by conservative diffeomorphisms on the surfaces. This is a joint work with X. Liu. Statistical properties and robust minimality for smooth random walks on surfaces | 24 January 2017 |

Stefano Marmi (Scuola Normale Superiore di Pisa) | **Abstract:** We introduce a coupled deterministic/slow - random/fast discrete time dynamical system to investigate the role of expectation feedbacks on systemic stability of financial markets. In our stylized world financial institutions are subject to value at risk constraints, follow standard mark-to-market and risk management rules and invest in some risky assets whose prices evolve stochastichally and are endogenously driven by the impact of portfolio decisions of financial institutions.When the traders become more myopic and build their expectations giving more weight to short term volatility estimates the equilibrium solution becomes unstable, leverage cycles arise and a cascade of bifurcations leading to chaos follows.
This is joint work with Fabrizio Lillo and Piero Mazzarisi. When Panic Makes You Blind: a Chaotic Route to Systemic Risk | 18 January 2017 |

Pierre Berger (CNRS-LAGA, Université Paris 13, UPSC) | **Abstract:** Recently we showed that some degenerate bifurcations can occur robustly. Such a phenomena enables ones to prove that some pathological dynamics are not negligible and even typical in the sens of Arnold-Kolmogorov. More precisely, we proved:
Theorem:
For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the mapping $f_a$ displays infinitely many sinks.
We will introduce the concept of Emergence which quantifies how wild is the dynamics from the statistical viewpoint, and we will conjecture the local typicality of super-polynomial ones in the space of differentiable dynamical systems.
For this end, we will develop the theory of Para-Dynamics, by giving a negative answer to the following problem of Arnold (1992):
Theorem:
For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the map $f_a$ displays a fast increasing number of periodic points:
$$\limsup \frac{\log Card \; Per_n \, f_a}n = \infty$$
We also give a negative answer to questions asked by Smale 1967, Bowen in 1978 and by Arnold in 1989, for manifolds of any dimension $\ge 2$:
Theorem:
For every $\infty\ge r\ge 1$, for every manifold of dimension $\ge 2$, there exists an open set $U$ of $C^r$-diffeomorphisms, so that a generic $f\in U$ displays a fast growth of the number of periodic points.
The proof involves a new object, the $\lambda$-$C^r$-parablender, the Renormalization for hetero-dimensional cycles, the Hirsh-Pugh-Shub theory, the parabolic renormalization for parameter family, and the KAM theory. Emergence and Para-Dynamics | 17 January 2017 |

Mark Pollicott (University of Warwick) | **Abstract:** TBA Analytic Cantor sets and stationary measures | 13 December 2016 |

Julian Newman (University of Bielefeld) | **Abstract:** Suppose we have a parameter-dependent orientation-preserving circle homeomorphism, together with a probability measure $\nu$ on the parameter space. This naturally generates a "random dynamical system" on the circle, where at each time step a parameter is randomly chosen with distribution $\nu$ (independently of all previous time steps) and the associated homeomorphism is applied. Given two parameter-dependent orientation-preserving homeomorphisms defined over the same parameter space (with the same measure), one can define a notion of "topological conjugacy" between the random dynamical systems that they generate. Under certain assumptions, we will classify such parameter-dependent homeomorphisms up to topological conjugacy. Topological conjugacy of iterated random orientation-preserving homeomorphisms of the circle | 13 December 2016 |

Raj Prasad (University of Massachussetts) | Multitowers, conjugacies and codes | 13 December 2016 |

Ricardo Pérez-Marco (CNRS, IMJ-PRG, Paris 7) | **Abstract:** We review the current knowledge on the structure of hedgehogs and its dynamics and present some of the the main open problems. Open problems in hedgehogs dynamics | 13 December 2016 |

André de Carvalho (USP) | **Abstract:** The purpose of this talk is not to set a new record for the number of noun modifiers in mathematical definitions, but to present a construction which applies to graph maps in general and yields interesting surface homeomorphisms as follows:
1) a pseudo-Anosov map if the graph map is a train track map;
2) a generalized pseudo-Anosov map if the graph map is post-critically finite and has an irreducible aperiodic transition matrix;
3) an interesting type of surface homeomorphisms which generalizes both the previous classes otherwise.
In particular, this produces a unified construction of surface homeomorphisms whose dynamics mimics that of the tent family of interval endomorphisms, completing an earlier construction of unimodal generalized pseudo-Anosov maps in the post-critically finite case.
This is joint work with Phil Boyland and Toby Hall. TBSpAgs: Super generalized pseudo-Anosov maps | 6 December 2016 |

Johan Taflin (Université de Bourgogne) | **Abstract:** In complex dynamics in several variables, the classical dichotomy between Julia and Fatou sets is insufficient to describe the dynamics. On the other hand, attractors are fondamental objets in the general theory of dynamical systems. In this talk we will explain how to start the ergodic study of attractors in the projective space using pluripotential theory. In particular, to each attractor we will associated an attracting current and an equilibrium measure which give geometric and dynamical information about the attractor. Currents associated to attractors in complex dynamics | 30 November 2016 |

Liviana Palmisano (University of Bristol) | **Abstract:** We prove that circle maps with a flat interval and degenerate geometry are an example of a dynamical system for which the topological classes don't coincide with the rigidity classes.
Contrarily to all the well-known examples in one-dimensional dynamics (such as circle diffeomorphisms, unimodal interval maps at the boundary of chaos, critical circle maps) we show that the class of functions with Fibonacci rotation numbers is a C^1 manifold which is foliated with finite codimension rigidity classes. This is a joint work with M. Martens. Foliations of Rigidity Classes | 29 November 2016 |

Sanju Velani
(University of York) | **Abstract:** The aim is to initiate a ``manifold'' theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural ``manifold'' strengthening of Sullivan's logarithmic law for geodesics. Diophantine approximation in Kleinian groups: singular, extremal and
bad limit points | 22 November 2016 |

Jimmy Tseng (University of Bristol) | **Abstract:** A theme in ergodic theory is exploring how much information ergodicity itself, without using stronger properties such as mixing, can give about an ergodic dynamical system. In this talk, we will focus on the system given by diagonal flows on the space of unimodular lattices and use the Birkhoff ergodic theorem, the Siegel mean value theorem, and an approximation argument to give an alternative proof of a form of Schmidt’s theorem on the number of integer lattice solutions of a system of inequalities where the solutions have uniformly bounded height. Joint work with J. Athreya and A. Parrish. Applications of the Birkhoff ergodic theorem and the Siegel mean value theorem to Diophantine approximation | 15 November 2016 |

Oleg Smolyanov (Moscow University) | Feynman path integrals and quantum anomalie | 10 November 2016 |

Toby Hall (University of Liverpool) | **Abstract:** In joint work with Phil Boyland and Andre de Carvalho we show that, for every unimodal map $f$ with topological entropy $h(f)\ge \frac{1}{2}\log2$, the natural extension of $f$ is semi-conjugate to a sphere homeomorphism. I will discuss some of the ideas of the proof and the dynamics of the resulting sphere homeomorphisms. Sphere homeomorphisms from unimodal maps | 8 November 2016 |

Vjacheslav Grines (Novgorod State University) | **Abstract:** This talk is devoted to finding sufficient conditions for the existence of closed and heteroclinic trajectories of Morse-Smale flows on a smooth closed orientable three-dimensional manifold $M^3$. Let $f^t$ Morse-Smale flow on $M^3$, whose non-wandering set consists of $k$ saddles and $ \ell $ nodes (sinks and sources). Set $g=\frac{k-\ell + 2}{2}$.
We represent two results:
1. There exists a Morse-Smale flow without closed and heteroclinic trajectories on
$M^3$ if and only if $M^3$ is the sphere $S^3$ and $k = \ell - 2$, or $M^3$ is the connected sum of $g$ copies of $S^2\times S^1$.
2. If the Heegaard genus of $ M ^3 $ more than the number $g$ then non-wandering set of a Morse-Smale flow $ f ^ t $ has at least
one closed trajectory. On inter-relation between existence of heteroclinic and closed trajectories of Morse-Smale flows and the topology of the ambient manifold. | 1 November 2016 |

Olga Pochinka (Novgorod State University) | **Abstract:** In this talk I will discuss the problem of topological classification of simple structural stable system, from Andronov-Potryagin to the present time. Classification of Morse-Smale dynamical systems | 1 November 2016 |

Nikolaos Karaliolios (Imperial College London) | **Abstract:** After having introduced some definitions, we will present a classification holding true in an open set of the total space of such dynamical systems. The phenomena that occur comprise the foliation of the phase space in smooth KAM tori, in tori of lower regularity down to simply measurable ones, and (for systems where the tori break down completely) Weak Mixing in the fibres and Unique Ergodicity in the space of Distributions. This classification is based on works of H. Eliasson, R. Krikorian and the author, among others.
The KAM regime for Quasi-Periodic cocycles in T^dxSU(2) | 25 October 2016 |

Laurent Stolovitch (Université de Nice-Sophia Antipolis) | **Abstract:** We consider germs of holomorphic vector fields at a fixed
point having a nilpotent linear part at that point, in dimension $n\geq
3$. Based on Belitskii's work, we know that such a vector field is
formally conjugate to a (formal) normal form. We give a condition on
that normal form which ensure that the normalizing transformation is
holomorphic at the fixed point. We shall show that this sufficient
condition is a {\it nilpotent version} of Bruno's condition (A). In
dimension 2, no condition is required since, according to
Str{\'o}{\.z}yna-{\.Z}o{\l}adek, each such germ is holomorphically
conjugate to a Takens normal form. Our proof is based on Newton's method
and $\frak{sl}_2(\Bbb C)$-representations. Holomorphic normal form of nonlinear perturbations of nilpotent vector fields | 13 October 2016 |

Erik Bollt (Clarkson University) | Identifying Interactions in Complex Networked Dynamical Systems through Causation Entropy | 14 September 2016 |

Isabel Rios (UFF) | Uniqueness of equilibrium states for a family of partially hyperbolic systems | 14 September 2016 |

Björn Winckler (Stony Brook University) | **Abstract:** Two infinitely renormalizable unimodal maps of bounded combinatorics which
belong to the same topological conjugacy class have Cantor attractors which on
small scales look the same. Colloquially, we say that topology determines
geometry. This is what we have come to expect for interval maps.
In this talk I will discuss recent surprising results that show that topology
does not always determine geometry in the case of bounded combinatorics Lorenz
maps. Topology determines geometry? | 23 August 2016 |

Rafael Obaya (University of Valladolid) | **Abstract:** In this talk the dissipativity of a family of non-autonomous
linear-quadratic control processes is studied using methods of
topological dynamics. The application of the Pontryaguin Maximum
Principle to this problem give rise to a family of linear Hamiltonian
systems for which the existence of an exponential dichotomy is assumed,
but no condition of controllability is imposed.
As a consequence, some of the systems of this family could be abnormal
and some of their dynamical properties are given. Sufficiente
conditions for the dissipativity of the processes are provided assuming
the existence of global positive solutions of the Riccati equation
induced by the family of linear hamiltonian sytems or by a conveniente
disconjugate perturbation of it. Abnormal linear hamiltonian systems with application in non-autonomous linear-quadratic control processes | 22 July 2016 |

Peter Hazard (University of Toronto) | **Abstract:** In the early 1980's it was observed that period-doubling cascades in families of area-preserving planar diffeomorphisms occur at a universal rate. As in the one-dimensional case it was conjectured that there exists a renormalization operator, acting on some class of area-preserving planar maps, possessing a hyperbolic fixed-point, with codimension-one stable manifold and one-dimensional unstable manifold. Later, a computer-assisted proof of the existence of the fixed point was given by Eckmann, Koch and Wittwer. However, a conceptual proof is still missing. I will discuss recent progress on this problem. This is joint work with D. Gaidashev. Period-Doubling Renormalization of Area-Preserving Planar Maps | 23 June 2016 |

Arnaldo Nogueira (Institut de Mathématiques de Marseille) | **Abstract:** Let I be the unit interval [0,1) and -1 < λ < 1. Let f : I → R be a piecewise λ-affine map, that is, there are real numbers b_1;,..., b_u and a sequence of points 0 = c_1 < c_2 <...< c_{u-1} < c_u = 1 such that f(x)= λx + b_i, for every x in the interval [c_{i-1} , c_i). In the talk, we examine the class of maps f_ρ = f + ρ mod 1, where ρ is a real parameter. We prove that, for Lebesgue almost every real parameter \rho, the map f_ρ is asymptotically periodic. More precisely, f_ρ has at most 2n periodic orbits and the ω-limit set of every x in the unit interval I is a periodic orbit. Our theorem is an extension of the result obtained for the much-studied case where λ is a positive constant and f is the continuous map x → λx . This is a joint work with Benito Pires and Rafael Rosales. Topological dynamics of piecewise λ-affine maps | 16 June 2016 |

Ale Jan Homburg (University of Amsterdam) | **Abstract:** I will discuss iterated function systems on the unit interval. The iterated function systems are generated by diffeomorphisms or by logistic maps. I will discuss possible dynamics such as intermittency in these contexts.
Random interval maps | 14 June 2016 |

Huaizhong Zhao (Loughborough University) | **Abstract:** TBA Ergodicity of Random Dynamical Systems Under Random Periodic Regimes
| 2 June 2016 |

Chunrong Feng (Loughborough University) | **Abstract:** TBA Random periodic solutions of SDEs | 2 June 2016 |

Tuomas Sahlsten (University of Bristol) | **Abstract:** Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière is an equidistribution result of eigenfunctions of the Laplacian in large frequency limit on a Riemannian manifold with an ergodic geodesic flow. We complement this work by introducing a Quantum Ergodicity theorem on hyperbolic surfaces, where instead of taking high frequency limits, we fix an interval of frequencies and vary the geometric parameters of the surface such as volume, injectivity radius and genus. In particular, we are interested in such results under Benjamini-Schramm convergence of hyperbolic surfaces.
This work is inspired by analogous results for holomorphic cusp forms and eigenfunctions for large regular graphs. The proof uses mixing properties of the geodesic flow together with a wave propagation approach recently considered by Brooks, Le Masson and Lindenstrauss on discrete graphs, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit, in the sense that it does not use any microlocal analysis. Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces | 26 May 2016 |

Esmerelda Sousa Dias (IST, Lisbon) | **Abstract:** Cluster maps are birational maps arising from mutation-periodic quivers. These maps preserve a log-canonical presymplectic form $\omega$ (defined in terms of a skew-symmetric matrix B which simultaneously defines the map) and so they can be reduced to symplectic maps. On the other hand, the existence of certain quadratic Poisson structures for which the cluster map is a Poisson map, leads to another reduction of the same map.
It will be explained how the null foliation of $\omega$, the symplectic foliation of the referred Poisson structures, and the dynamics of the reduced maps, offer a complete understanding of the geometry underlying the (discrete) dynamics of some cluster maps.
This is joint work with Inês Cruz (FCUP) and Helena Mena-Matos (FCUP) Poisson structures and the dynamics of cluster maps | 24 May 2016 |

Jérémy Blanc (Universität Basel) | **Abstract:** "One way to study the iterations or birational maps is to compute the sequence of degrees we obtain. The type of growth ( bounded / polynomial / exponential) is well-known in dimension 2 and is useful to study the dynamics of the map (entropy, growth of fixed points of iterates,...) but also the conjugacy class, the centraliser, … I will describe the results one knows in this direction and some of the open questions in higher dimension." Degree growth of birational maps | 19 May 2016 |

Davoud Cheraghi (Imperial College London) | **Abstract:** In the 1970’s, physicists, working numerically, observed “universal scaling laws” in the bifurcation loci of generic families of one-dimensional analytic transformations. To explain this phenomena, they conjectured that a non-linear (renormalisation) operator acting on an infinite-dimensional function-space is hyperbolic which has a one-dimensional unstable direction and a co-dimension-one stable direction. This was the focus of research in the 90’s, leading to a rigorous proof of the conjecture on subspaces where the operator is compact, while a successful study of the problem on subspaces where the operator is not compact has been obtained only recently. In an introductory talk, we present the motivations and some of the key ideas employed in the subject. Hyperbolicity of renormalization operators | 6 May 2016 |

Stefano Luzzatto (ICTP) | SRB measures for nonuniformly hyperbolic surface diffeomorphisms | 19 April 2016 |

John Smillie (Univeristy of Warwick) | **Abstract:** I will begin this talk with a general introduction to the dynamics of
the complex Henon family of diffeomorphisms. What might we learn by studying
them? What techniques have proven effective? I will discuss joint work with
Eric Bedford which addresses the question of what is the natural two dimensional
analogue of the Misiurewicz property for polynomial maps. This work addresses
the connection between regularity of stable and unstable manifolds and uniformity
of expansion and the presence of tangence’s. Notions of regularity for Complex Henon Diffeomorphisms | 17 March 2016 |

Dalia Terhesiu (University of Exeter) | **Abstract:** For non-uniformly expanding maps inducing w.r.t. a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining sharp estimates for the correlation function. This applies to both, finite and infinite measure setting. The results are illustrated by non Markov interval maps with an indifferent fixed point. This is joint work with H. Bruin. Sharp mixing rates via inducing w.r.t. general return times | 17 March 2016 |

Stefano Marmi (Scuola Normale Superiore di Pisa) | Regularity of solutions of the cohomological equation for interval exchange maps | 11 March 2016 |

Katsutoshi Shinohara (Hitotsubashi University) | **Abstract:** beta-encoders are Analog-to-Digital encoders based on
beta-expansions. In order to give estimates of their quantization errors,
the analysis of corresponding piecewise-linear maps plays an
important role. In this talk, I will talk about this interplay between
electronic circuit design theory and dynamical systems. beta-encoders and Fredholm determinant of generalized beta-maps | 11 March 2016 |

Henna Koivusalo (University of York) | **Abstract:** We study local properties of fractal sets, their tangent sets. These are the limiting patterns in the Hausdorff metric when zooming in to a point, along a sequence of scales converging to 0. Tangent sets give a good description of the fine structure of the fractal set, and are well-understood for self similar sets. We investigate tangent sets of self affine sets in the plane. We prove that under some natural assumptions on the self affine set, almost everywhere the tangent sets have a fibered structure; that is, they are the product of a line segment with a Cantor set in a suitably chosen basis. This is in stark contrast to the self similar case, where the tangent sets are deformations of the self-similar set itself. Self affine sets with a fibered tangent structure | 10 March 2016 |

John Mackay (University of Bristol) | **Abstract:** Groups can be investigated by considering how they can act on suitable
spaces. For example, the notion of Kazhdan's property (T), relating to
how groups can act on Hilbert spaces, has been used very successfully
for many applications over the last fifty years. More recently, similar
definitions have been used to study actions on other L^p spaces, with
motivations from dynamics amongst other fields. After outlining some of
this story, I'll explain why actions of certain random groups on L^p
spaces have fixed points. (Joint work with Cornelia Drutu. Fixed point properties for groups acting on L^p spaces | 3 March 2016 |

Paul Verschueren (Open University) | **Abstract:** We discuss an important class of functions, denoted Quasiperiodic Sums and Products, which link the study of critical phenomena in diverse fields such as the birth of Strange Non-Chaotic Attractors, Critical KAM Theory, and q-series (much used in String Theory). They also link study of pure topics in Power Series analysis, Partition Theory, and Diophantine Approximation.
The graphs of these functions form intriguing geometrically strange and self-similar structures. They are easy and rewarding to investigate numerically, and suggest many avenues for investigation. However they prove remarkably resistant to rigorous analysis.
In this talk we will review results on the most heavily studied example, Sudler's product of sines. We will also report on our own new work towards developing a rigorous theoretical foundation in this area. In particular this allows us to settle negatively a conjecture of Erd?s & Szekeres from 1959, and to prove a number of experimental results reported recently by Knill & Tangerman (2011). Quasiperiodic sums and products | 2 March 2016 |

Charles Walkden (University of Manchester) | **Abstract:** We define and explore the notion of finitely coupled iterated function system that satisfy a notion of contraction on average. We study the existence and uniqueness of an invariant
probability measure for such systems by defining an appropriate transfer operator. Using arguments of Keller and Liverani on properties of perturbations of linear operators, the continuity of the invariant measure as the coupling tends to zero is proved. Other limit theorems, including a central limit theorem, can also be proved. This is joint work with Anthony Chiu. Stochastic stability and limit theorems for coupled iterated function systems that contract on average
| 25 February 2016 |

Neil Dobbs (University of Geneva) | **Abstract:** Free energy is not semi-continuous, but we show that in some contexts, it nearly is. This engenders proofs of existence of equilibrium states and (almost) continuity of equilibrium states as one varies the potential and the map. We do this in the context of one-dimensional dynamics. Joint with M. Todd. Free energy jumps up | 22 February 2016 |

Masayuki Asaoka (University of Kyoto) | A C-infinity closing lemma for Hamiltonian diffeomorphisms on surfaces | 10 February 2016 |

Péter Bálint (Budapest University of Technology and Economics ) | **Abstract:** TBA Mean field coupling of doubling maps | 5 February 2016 |

Stergios Antonakoudis (Cambridge University) | **Abstract:** TBA The complex geometry of Teichmüller spaces and bounded symmetric domains | 4 February 2016 |

Oleg Smolyanov (Moscow University) | Quasi-invariant measures on infinite-dimensional spaces | 4 February 2016 |

Diana Tolstyga (Moscow University) | Feynman formulas for stochastic and quantum dynamics of particles in multidimensional domains | 3 February 2016 |

Oleg Ivrii (University of Helsinki) | **Abstract:** We examine several characteristics of conformal maps that resemble the
variance of
a Gaussian: (1) asymptotic variance, (2) the constant in Makarov’s law
of iterated
logarithm and (3) the second derivative of the integral means spectrum
at the
origin, amongst others. While these quantities need not be equal in
general, they
agree for domains whose boundaries are regular fractals such as Julia
sets or
limit sets of quasi-Fuchsian groups. We show these characteristics have
the same
universal bounds over various collections of conformal maps. As an
application,
we show that the maximal Hausdorff dimension of a k-quasicircle is
strictly less
than 1 + k^2. (Part of this work is joint with I. Kayumov.) On Makarov’s principle in conformal mapping | 28 January 2016 |

Tiago Pereira (University of Sao Paulo, Sao Carlos) | **Abstract:** TBA Improving Network connectivity can lead to Functional Failures | 28 January 2016 |

Fabrizio Bianchi (University of Toulouse) | **Abstract:** Starting from the basics in holomorphic dynamics in one variable, I will review the classical theory by Mané-Sad-Sullivan about stability of rational maps and briefly present a generalization of this theory in higher dimension.
I will focus on the arguments that do not readily generalize in the second case, and introduce the tools and ideas that allow to overcome these problems. Holomorphic motions of Julia sets | 13 January 2016 |

Dmitry Todorov (Centre de Physique Théorique (Aix-Marseille Université)) | **Abstract:** There is a strong and long-lasting interest in chaotic dynamical systems
as mathematical models of various processes in different areas of science.
Like for any other mathematical models for chaotic systems to be useful it
is desirable that they have stability properties.
There are exist different stability properties. In particular, there exist
two notions of stability with respect to small per-iteration perturbations –
shadowing property and stochastic stability.
System is said to have shadowing property if every (pseudo)trajectory
with small errors can be uniformly approximated by a trajectory without
errors. System is stochastically stable if the noise perturbing the system is
considered to be random and invariant measures for the stochastic process
corresponding to the perturbed system converge to an invariant measure of
the unperturbed system.
Although conceptually these properties are somewhat similar and it is
known that some chaotic systems have both properties, no direct relations
between shadowing and stochastic stability were established so far.
I will discuss some of these relations both for qualitative and quantitative
versions of shadowing and stochastic stability. Shadowing and stochastic stability | 10 December 2015 |

Anton Solomko (University of Bristol) | **Abstract:** A probability preserving action T is called simple if the set of its self-joinings is the smallest possible and arise from the centralizer C(T) of T. For simple actions, there is an interplay between algebraic properties of the group C(T) and ergodic properties of T.
First I will explain (C,F)-construction of measure preserving actions, which is an algebraic counterpart of classical cutting-and-stacking technique. This method combined with Ornstein's technique of 'random spacers' allows to produce simple actions with centralizers prescribed in advance.
Then I will focus on some new examples, obtained via (C,F)-construction, including mixing actions with uncountably many prime factors and mixing transformations with infinitely many non isomorphic embeddings into a flow.
Based on joint works with Alexandre Danilenko and Joanna Ku?aga-Przymus. On centralizers of simple mixing actions | 10 December 2015 |

Caroline Series (University of Warwick) | **Abstract:** A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Its limit set, contained in the Riemann sphere, is the set of accumulation points of any orbit. In particular the limit set of a hyperbolic surface group F is the unit circle.
If G is a Kleinian group abstractly isomorphic to F, there is an induced map, known as a Cannon-Thurston (CT) map, between their limit sets. More precisely, the CT-map is a continuous equivariant map from the unit circle into the Riemann sphere.
Suppose now F is fixed while G varies. We discuss work with Mahan Mj about the behaviour of the corresponding CT-maps, viewed as maps from the circle to the sphere. We explain how a simple criterion for the existence of a CT-map can be adapted to establish conditions on convergence of a sequence of groups G_n under which the corresponding sequence of CT-maps converges uniformly to the expected limit. We also discuss an example which shows that under certain circumstances, CT-maps may not even converge pointwise.
Continuous motions of limit sets | 26 November 2015 |

David Simmons (University of York) | **Abstract:** We consider a class of measures from Diophantine approximation known as \emph{extremal} measures. The class of measures known to be extremal has expanded in recent years to include not only the Lebesgue measures of nondegenerate manifolds, but also various measures defined using conformal dynamics. In this talk I will describe this history as well as describing a new geometric condition which implies extremality, giving examples of dynamical measures satisfying this condition which could not previously be proven to be extremal. This work is joint with Tushar Das, Lior Fishman, and Mariusz Urba?ski. Extremality and dynamically defined measures | 19 November 2015 |

Simon Baker (University of Reading) | **Abstract:** Expansions in non-integer bases is a natural generalisation of the well know binary/tertiary/decimal expansions. Associated to these expansions is a natural class of dynamical systems. In this talk I will introduce these dynamical systems and discuss their first return dynamics. In particular, I will discuss allowable sequences of return times, and when the first return map is a generalised Luroth series transformation. Induced random beta transformations
| 12 November 2015 |

Zemer Kosloff (University of Warwick) | **Abstract:** Markov partitions introduced by Sinai and Adler and Weiss
are a tool that enables transfering questions about ergodic theory
of Anosov Diffeomorphisms into questions about Topological Markov
Shifts and Markov Chains. This talk will be about a reverse reasonning,
that gives a construction of $C^{1}$ conservative (satisfy Poincare\textquoteright s
reccurrence) Anosov Diffeomorphism of $\mathbb{T}^{2}$ without a
Lebesgue absolutely continuous invariant measure. By a theorem of
Gurevic and Oseledec, this can\textquoteright t happen if the map
is $C^{1+\alpha}$ with $\alpha>0$. Our method relies on first choosing
a nice Toral Automorphism with a nice Markov partition and then constructing
bad conservative Markov measure on the symbolic space given by the
Markov partition. We then push this measure back to the Torus to obtain
a bad measure for the Toral automorphism. The final stage is to find
by smooth realization a conjugating map $H$ such that $H\circ f\circ H^{-1}$
with Lebesgue measure is metric equivalent to $\left(\mathbb{T}^{2},\mathcal{B},f,\ Bad\ measure\right)$. Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure | 5 November 2015 |

Carlos Siqueira (Imperial College London) | **Abstract:** We introduce for the first time the notion of hyperbolicity and structural stability for the one parameter family of (multi-valued) complex maps f_c(z) =z^r +c, where r > 1 is a rational number. Such a family is particularly important not only as a generalisation of the quadratic family but also because it is a conformal extension (as a Riemann surface) of some unimodal maps. We show that its Julia set is the projection of a solenoid and consists of uncountably many quasi-conformal arcs for parameters close to the origin. We also estimate the Hausdorff dimension of the Julia set using the formalism of Gibbs states.
The general structural stability is proved for every hyperbolic parameter using holomorphic motions on Banach spaces. For such parameters the limit set (accumulation points out of pre-orbits starting at the basin of infinity) splits into two disjoint, compact and invariant sets: the standard Julia set and the dual Julia set. The former is the closure of repelling periodic orbits and the latter is the closure of attracting periodic orbits. Surprisingly, the dual Julia set is typically a Cantor set associated with a Conformal Iterated Function System. Therefore, we have infinitely many attracting periodic points! This set is always finite for rational maps. Cantors sets and structural stability for holomorphic correspondences | 29 October 2015 |

Alexey Korepanov (University of Warwick) | **Abstract:** We consider a family of Pomeau-Manneville type interval maps $T_\alpha$,
parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant
probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$.
We show that despite the absence of a spectral gap for all $\alpha \in (0,1)$ and
despite nonsummable correlations for $\alpha \geq 1/2$, the map
$\alpha \mapsto \int \varphi \, d\nu_\alpha$ is continuously differentiable
for $\varphi \in L^{q}[0,1]$ for $q$ sufficiently large. Linear response for intermittent maps with summable and nonsummable decay of correlations | 22 October 2015 |

Bastien Fernandez (Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
CNRS - Université Paris 7 Diderot) | **Abstract:** The Kuramoto model is the archetype of heterogeneous systems of (globally) coupled oscillators with dissipative dynamics. In this model, the order parameter that quantifies the population synchrony decays to 0 in time, as long as the interaction strength remains small (so that the uniformly distributed stationary solution remains stable).
While this phenomenon has been identified since the first studies of the model, its proof remained to be provided (most studies in the literature are limited to the linearized dynamics).
The goal of this talk is to address this issue. I will present rigorous results on the Kuramoto dynamics, and in particular, I will sketch a proof of nonlinear damping of the order parameter in the incoherent phase.
Joint work with D. Gérard-Varet and G. Giacomin Landau damping in the Kuramoto model | 8 October 2015 |

Sina Tureli (ICTP) | **Abstract:** We are going to give a new theorem for integrability of rank 2 continuous tangent sub-bundles (distributions) in 3 dimensional manifolds. The theorem depends on approximations and a natural notion of asymptotic involutivity. We will then state some applications to PDE, ODE and Dynamical Systems. A Frobenius Theorem for Continuous Distributions | 2 September 2015 |

Gary Froyland (University of New South Wales) | **Abstract:** Transfer operators are global descriptors of ensemble evolution under nonlinear dynamics and form the basis of efficient methods of computing a variety of statistical quantities and geometric objects associated with the dynamics.
I will discuss two related methods of identifying and tracking coherent structures in time-dependent fluid flow; one based on probabilistic ideas and the other on geometric ideas.
Applications to geophysical fluid flow will be presented. Transfer operators and dynamics | 23 July 2015 |

Christian Kühn (Technical University of Vienna) | **Abstract:** In this talk I am going to report on the geometric decomposition of
nonlinear dynamics in the Olsen model. Although this model has been proposed
by Olsen already in the late 1970s and has been investigated many times with
di fferent methods, a full understanding of the mechanisms that lead to
oscillatory patterns was not available.
Nonlinearity, several small parameters, higher-dimensionality and wide
parameter ranges are the key difficulties in this context. However, using
methods from the geometric theory of multiple time scale dynamical systems,
it is possible to identify the main mechanisms. In particular, I am going to
illustrate the main steps to prove the existence of non-classical relaxation
oscillations and explain how one may deal with mixed-modes and chaotic
solutions from the same viewpoint. Multiscale Oscillations in the Olsen Model | 16 July 2015 |

Arno Berger (University of Alberta) | **Abstract:** The study of numbers generated in one way or another by dynamical
systems is a classical, multifaceted field. A notorious gem in this
field is the wide-spread, unexpected emergence of a particular
logarithmic distribution, commonly referred to as Benford's Law (BL).
This talk will describe how dynamical systems may conform to BL,
and what this in turn may tell about the dynamics in question. As
one illustrative example, a characterization of BL in finite-dimensional
linear systems, recently obtained in joint work with G. Eshun, will be
discussed in some detail. While this result is quite general and implies,
for instance, that such systems typically conform to BL in a very strong
sense, it also raises intriguing new questions.
Digit distributions in dynamics | 25 June 2015 |

Ken Palmer (Providence University, Taiwan) | **Abstract:** This is joint work with Kaijen Cheng. There are two parts. In the first part,
we study unimodal maps on the closed unit interval,
which have a stable period 3 orbit and an unstable period 3 orbit,
and give conditions under which all points in the open unit interval
are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is chaotic.
For the particular value
of $\mu=3.839$, Devaney, following ideas of Smale and Williams, shows
that the logistic map $f(x)=\mu x(1-x)$ has this property.
In this case the stable and unstable period 3 orbits appear when $\mu=1+\sqrt{8}$.
We use our theorem to show that the property holds for all values of $\mu>1+\sqrt{8}$
for which the stable period 3 orbit remains stable.
\medskip
In the second part we study a model for masting, that is,
the intermittent production of flowers and fruit by trees.
This model is an asymmetric unimodal piecewise linear map
with one parameter $k>0$. When $k<1$ all orbits are attracted to a stable fixed point. On the other hand, when $k>\kappa_0=(1+\sqrt{5})/2$, the map is chaotic on $[1-k,1]$.
As $k$ increases through $1$, chaos arises immediately.
We find a strictly decreasing sequence $\kappa_p$, $p\ge 0$, with
$\kappa_p\to 1$ as $p\to\infty$ such that when $\kappa_p <= k < \kappa_{p-1}$, $p\ge 1$,
$g$ has an invariant set $\Lambda(k,p)$ consisting
of $2^p$ disjoint closed intervals on which the dynamics is chaotic and
to which all points except for a countable set are attracted;
as $k$ decreases through
$\kappa_p$ each of the $2^p$ intervals splits into two (the middle part drops out).
$\Lambda(k,p)$ is squeezed towards the two point set
$\{0,1\}$ as $p\to\infty$. Period 3 and Chaos for Unimodal Maps | 18 June 2015 |

Olga Pochinka (Novgorod State University) | **Abstract:** http://wwwf.imperial.ac.uk/~tclark/Pochinka-abstract.pdf On topological classification of diffeomorphisms of surfaces with a finite number of moduli of stability | 5 June 2015 |

Vjacheslav Grines (Novgorod State University) | **Abstract:** http://wwwf.imperial.ac.uk/~tclark/Grines-abstract.pdf On topological classification of structurally stable cascades with two-dimensional basic sets on 3-manifolds | 5 June 2015 |

Thomas Kaijser (Linköping University) | **Abstract:** I shall consider three examples: 1) Stochastic perturbations of
iterations of an analytic homeomorphism of the circle having
irrational rotation number; 2) Stochastic iterations of
the two functions f(x)= ax(1-x) and g(x) = bx(1-x);
and 3) Stochastic perturbation of iterations of the function
f(x)= x/2 +17/30 (mod 1). I will mainly spent time on the first example. On stochastic iterations of circle maps and interval maps | 28 May 2015 |

Alexandre De Zotti (University of Liverpool) | **Abstract:** We recall some results on the rigidity of dynamics of transcendental entire functions in the class B near infinity, obtained by Lasse Rempe-Gillen and Gwyneth Stallard. Those results are based on quasiconformal rigidity of the dynamics of class B functions near infinity. Different dimension quantities are known to be invariant in affine equivalence classes of maps. Poincaré maps shows that this invariance may not hold for quasiconformal classes. This is a joint work with Lasse Rempe-Gillen. The eventual hyperbolic dimension of entire functions | 14 May 2015 |

Giancarlo Benettin (University of Padova) | **Abstract:** In 1954, Fermi, Pasta and Ulam for the first time used a
computer to understand the ergodic behavior of a dynamical system with
many degrees of freedom, interesting to investigate the very
foundations of Statistical Mechanics. Several baranches of physical
and mathematical investigations started from that paper. The aim of
the talk is to revisit, in the light of some recent numerical results,
some significant ideas and conjectures on the model. In particular:
(i) The presence, in the model, of (at least) two well separated
time-scales: a short one, where only a few normal coordinates share energy,
and a larger one, where energy equipartition among all normal modes
occurs and the behavior of the model, in view of Statistical
Mechanics, is regular.
(ii) The fact that in the short time scale the dynamics of FPU, in
spite of the partial energy sharing, is essentially integrable and
closely follows the dynamics of the Toda model, while in
the large time scale nonintegrability becomes manifest.
The stability of results in the limit of large N (ideally, the search
of uniformity in N) will play a central role. The gap between
numerical insights and mathematical results, unfortunately, is large. The Fermi-Pasta-Ulam problem: old ideas, recent results, open problems | 11 May 2015 |

Christian Bick (University of Exeter) | **Abstract:** Phase oscillators are interacting oscillatory units whose state is solely determined by a phase variable taking values on the circle. Probably the most widely studied system of interacting phase oscillators is the Kuramoto model; here, the interaction between two oscillators is given by the sine of the phase difference. What happens if the interaction between pairs of oscillators is more general, for example if the interaction contains more than a single Fourier component? We discuss the impact of generalized coupling on phase oscillator dynamics. Moreover, we show how generalized couplings can be useful in applications; for example, it allows to control spatially localized states in non-locally coupled oscillator systems. Dynamics of Coupled Phase Oscillators with Generalized Coupling | 30 April 2015 |

Michael Tsiflakos (University of Vienna) | **Abstract:** TBA Chaotic behaviour of bouncing balls | 20 April 2015 |

Christian Pötzsche (Alpen Adria University of Klagenfurt) | **Abstract:** The dichotomy spectrum (also known as dynamical or Sacker-Sell spectrum) is a crucial notion in the theory of dynamical systems. It contains information on stability and robustness properties, and is moreover fundamental to establish a geometric theory including invariant manifolds, linearization and normal forms.
However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous equations to the (approximate) point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which
(a) allow to classify nonautonomous bifurcations on a linear basis already
(b) simplifies proofs for results on the long term dynamics of difference and differential equations with explicitly time-dependent right-hand side
(c) yield sufficient conditions for a continuous (rather than merely an upper-semicontinuous) behavior of the dichotomy spectrum under perturbation On the dichotomy spectrum | 9 April 2015 |

Dierk Schleicher (Jacobs University) | **Abstract:** We all know that Newton’s method is efficient as a local root finder, but has a reputation of being globally unpredictable. We present an algorithm that makes it globally efficient and predictable, and (news of a few days ago) make it possible to find, in practice, all roots of polynomials of degree one million in just a few seconds on a standard personal computer. (Party joint work with Robin Stoll.) Newton’s method as a practical root finder | 30 March 2015 |

Jonathan Fraser (University of Manchester) | **Abstract:** I will consider the abstract problem of ‘zooming in’ on a shift ergodic measure. This is motivated by recent and influential developments in the ergodic theory of the so called ‘scenery flow’, largely due to Mike Hochman and his collaborators. In order to capture the zoom in dynamics one defines a sequence of ‘scenery distributions’, which are measures on the space of measures defined by summing up Dirac masses along the orbit of the original measure under the zooming in process. I will show that at almost every point the scenery distributions converge to a common distribution on the space of measures and show how to characterise this distribution in terms of an appropriately defined reverse Jacobian.
This is based on joint work with Mark Pollicott. Blowing up ergodic measures | 19 March 2015 |

Vito Latora (Queen Mary University of London) | **Abstract:** TBA The growth of cities and neural networks | 19 March 2015 |

Liviana Palmisano (Institute of Mathematics Polish Academy of Sciences) | **Abstract:** We study C^2 weakly order preserving circle maps with a flat interval. We prove that, if the rotation number is of bounded type, then there is a sharp transition from the degenerate geometry to the bounded geometry depending on the degree of the singularities at the boundary of the flat interval.
The general case of functions with rotation number of unbounded type is also studied. The situation becomes more complicated due to the presence of underlying parabolic phenomena.
Moreover, the results obtained for circle maps allow us to study the dynamics of Cherry flows. In particular we analyze their metric, ergodic and topological properties. On circle endomorphisms with a flat interval and Cherry flows | 13 March 2015 |

Shin Kiriki (Tokai University) | **Abstract:** We give an answer to a version of the open problem of F. Takens 2008 which is related to historic behavior of dynamical systems. To obtain the answer, we show the existence of non-trivial wandering domains near a homoclinic tangency, which is conjectured by Colli and Vargas 2001. Concretely speaking, it is proved that any Newhouse open set in the C^r topology of two-dimensional diffeomorphisms with 2 \leq r < \infty is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior.
Moreover, this result implies an answer in the C^r category to one of the open problems of van Strien 2010 which is concerned with wandering domains for H\'enon family. Takens' last problem and existence of non-trivial wandering domains | 12 March 2015 |

Zin Arai (Hokkaido University) | **Abstract:** We discuss the structure of the parameter space of the Henon family. Our main
tool is the monodromy representation that assigns an automorphism of the full
shift to each loop in the hyperbolic parameter locus of the complex Henon
family. We show that the monodromy carries the information of the bifurcations
taking place inside the loop, and this enables us to construct pruning fronts,
a generalization of kneading theory to the real Henon family. Furthermore,
assuming that there exist infinitely many non-Wieferich prime numbers (it
suffices to assume "abc conjecture"), we show that monodromy automorphisms
must satisfy a certain algebraic condition, which imposes geometric
restrictions on the structure of the parameter space. On the monodromy and bifurcations of the Henon map | 12 March 2015 |

Remus Radu (Stony Brook University) | **Abstract:** We study the global dynamics of complex Hénon maps with a semi-parabolic fixed point that arise as small perturbations of a quadratic polynomial p with a parabolic fixed point. We prove that this family of semi-parabolic Hénon maps is structurally stable on the sets J and J^+; the Julia set J is homeomorphic to a quotiented solenoid (hence connected), while the Julia set J^+ inside a polydisk is a fiber bundle over the Julia set of the polynomial p. We then exhibit certain paths in parameter space for which the semi-parabolic structure can be deformed into a hyperbolic structure, and show that the parametric region of semi-parabolic Hénon maps with small Jacobian lies in the boundary of a hyperbolic component of the Hénon connectedness locus. This is joint work with Raluca Tanase. Semi-parabolic Hénon maps and perturbations | 12 March 2015 |

Peter Ashwin (University of Exeter) | **Abstract:** A chimera state in a coupled oscillator system is a dynamical state that combines regions of coherence (or synchrony) with regions if incoherence (or asynchrony). However exactly how one defines these terms affects which states can be identified as chimeras and which not, especially for small groups of phase oscillators. In this talk I will discuss some joint work with O. Burylko (Kiev) where propose a definition of a weak chimera based on partial frequency synchrony. This allows one to explore the existence and stability of weak chimeras in small networks of phase oscillators. In particular we find that the usual coupling considered when investigating chimeras (Kuramoto-Sakaguchi type) leads to rather degenerate sets of neutrally stable chimeras, while more generic coupling unfolds this degeneracy. Chimera states for minimal systems of phase oscillators | 5 March 2015 |

Jochen Bröcker (University of Reading) | **Abstract:** The filtering process is the conditional probability of the state of a Markov process (the signal process), given a series of observations which are conditionally independent given the signal process. Stability means that the distance between the true filtering process and a wrongly initialised filter converges to zero (at an exponential rate in our case) as time progresses. In the present setting, the signal process arises through iterating an iid series of random and uniformly expanding maps on a Riemannian manifold.
We will see that the problem bears a strong similarity to the problem of constructing random invariant measures for these mappings, and hence the connection with dynamical systems. A fruitful approach in both cases is to ensure that the Frobenius Perron operator (or the filtering operator) is a contraction wrt to Hilbert's projective metric on suitably defined cones of functions.
Stability of the nonlinear filter for random expanding maps | 26 February 2015 |

Rafael Labarca (Universidad de Santiago de Chile) | **Abstract:** TBA Bubbles of entropy in the Lexicographical world and applications to the quadratic family of Lorenz maps | 19 February 2015 |

Paul Glendinning (University of Manchester) | **Abstract:** TBA Dimension and geometry for piecewise smooth maps | 12 February 2015 |

Oscar Bandtlow (Queen Mary University of London) | **Abstract:** In a seminal paper Ruelle showed that the long time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of
the eigenvalues of a certain operator acting on a suitable Banach space of holomorphic functions. Ruelle also showed that these eigenvalues, also known as Pollicott-Ruelle resonances, are bounded from above by a decaying exponential.
In this talk I will focus on lower bounds for the Ruelle eigenvalues. More precisely, I will explain how to prove that there exists a dense set of analytic expanding circle maps for which the Ruelle eigenvalues enjoy exponential lower bounds.
This is based on work with W. Just, J. Slipantschuk, and F. Naud. Lower bounds for the Ruelle eigenvalues of analytic circle maps | 5 February 2015 |

Mike Field (Imperial College London) | **Abstract:** The talk is intended to be a general (and gentle) introduction
to models of network dynamics that are applicable to contemporary
problems in biology, engineering and technology. In particular,
we will discuss the limitations of classical models and how to deal with networks
where, for example, connection structure may vary in time and nodes may stop
and later restart. Asynchronous networks and event driven dynamics | 29 January 2015 |

Alex Clark (University of Leicester) | **Abstract:** TBA An invitation to the Pisot Conjecture | 29 January 2015 |

Wael Bahsoun (Loughborough University) | **Abstract:** We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate. Decay of correlation for random intermittent maps | 22 January 2015 |

Bruno Ziliotto (Toulouse School of Economics) | **Abstract:** A stochastic game is described by a set of states, an action set for each player, a payoff function and a transition function. At each stage, the two players simultaneously choose an action, and receive a payoff determined by the actions and the current state. Then a new state is drawn from a distribution depending on the actions and on the former state. We consider stochastic games with long duration, and investigate the existence of a concept of long-term equilibrium payoff, called the asymptotic value. A counterexample to a long-standing conjecture concerning the existence of the asymptotic value will be presented.
Knowledge of game theory is no prerequisite for this talk. All the basic concepts will be redefined and illustrated by examples. Asymptotic Value in Two-Player Zero-Sum Stochastic Games and the Mertens conjecture | 15 January 2015 |

Peter Giesl (University of Sussex) | **Abstract:** In this talk we consider two methods of determining the basin of attraction.
In the first part, we discuss the construction of a Lyapunov function using Radial Basis Functions. The basin of attraction of equilibria or periodic orbits of an ODE can be determined through sublevel set of a Lyapunov function. To construct such a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the ODE, a linear PDE is solved approximately using Radial Basis Functions. Error estimates ensure that the approximation is a Lyapunov function.
For the construction of a Lyapunov function it is necessary to know the position of the equilibrium or periodic orbit. A different method to analyse the basin of attraction of a periodic orbit without knowledge of its position uses a contraction metric and is discussed in the second part of the talk. A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. In this talk, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine function, which is affine on each simplex of a triangulation of the phase space. Construction of Lyapunov functions and Contraction Metrics to determine the Basin of Attraction | 15 December 2014 |

Mark Holland (University of Exeter) | **Abstract:** For dynamical systems we discuss the statistics of extremes, namely the statistical limit laws that govern the process $M_{n}=\max\{X_1,X_2,\ldots,X_n\}$ , where $X_i$ correspond to a stationary time series of observations generated by the dynamical system. We discuss extreme statistics for a range of examples of interest to those working in ergodic theory and chaotic dynamical systems. In a work in progress, we discuss almost sure growth rates of $M_n$, and the statistics of records: namely the distribution of times $n$ such that $X_{n}=M_n$. On Extremes, recurrence and record events in dynamical systems | 11 December 2014 |

Mike Todd (University of St. Andrews) | **Abstract:** Fernandez and Demers studied the statistical properties of the Manneville-Pomeau map with the physical measure when a hole is put in the system, overcoming some of the problems caused by subexponential mixing. I’ll discuss the same setup, but with a class of natural equilibrium states. We find conditionally invariant measures and give precise information on the transitions between the fast exponentially mixing, the slow exponentially mixing and the subexponentially mixing phases. This is joint work with Mark Demers. Dynamical systems with holes: slow mixing cases | 11 December 2014 |

Gabriel Paternain (University of Cambridge) | **Abstract:** I will discuss the inverse problem of recovering a unitary connection from
the parallel transport along geodesics of a compact Riemannian manifold of negative curvature
and strictly convex boundary. The solution to this geometric inverse problem is based on a range of techniques,
including energy estimates and regularity results for the transport equation associated with the
geodesic flow.
Recovering a connection from parallel transport along geodesics | 5 December 2014 |

Han Peters (Universiteit van Amsterdam) | **Abstract:** Sullivan's non-wandering domains theorem from 1985 showed that rational functions do not have wandering Fatou components, which completed the classification of Fatou components in the Riemann sphere. In this talk I will discuss recent work with Matthieu Astorg, Xavier Buff, Romain Dujarin and Jasmin Raissy showing that there exist polynomial maps in two variables with wandering Fatou components. While our methods are complex, we also obtain real polynomial maps with wandering domains. The main idea, suggested by Misha Lyubich, is to apply parabolic implosion techniques to polynomial skew products. A polynomial map in two variables with a wandering Fatou component | 5 December 2014 |

Nikita Sidorov (University of Manchester) | **Abstract:** In this talk I will consider a natural two-parameter family of self-affine iterated function systems in the plane and provide a detailed analysis of their attractors. In particular, I will describe new results on the set of parameters for which the attractor has a non-empty interior and on the connectedness locus for this family. I will also talk about the set of uniqueness and simultaneous signed beta-expansions. This talk is based on a joint paper with Kevin Hare. On a family of two-dimensional self-affine sets | 27 November 2014 |

Nadia Sidorova (University College London) | **Abstract:** The parabolic Anderson problem is the Cauchy problem for the heat equation
on the d-dimensional integer lattice with random potential. It describes the behaviour
of branching random walks in a random environment (represented by the potential)
and is being actively studied by mathematical physicists. One of the most important
situations is when the potential is time-independent and is a collection of independent
identically distributed random variables. We discuss the intermittency effect occurring
for such potentials and consisting in increasing localisation and randomisation of the
solution. We also discuss the ageing behaviour of the model showing that the periods,
in which the profile of the solutions remains nearly constant, are increasing linearly
over time. Localisation and ageing in the parabolic Anderson model | 20 November 2014 |

James Meiss (University of Colorado Boulder) | **Abstract:** Turbulent fluid flows have long been recognized as a superior means of diluting initial concentrations of scalars due to rapid stirring. Conversely, experiments have shown that the structures responsible for this rapid dilution can also aggregate initially distant reactive scalars and thereby greatly enhance reaction rates. Indeed, chaotic flows not only enhance dilution by shearing and stretching, but also organize initially distant scalars along transiently attracting regions in the flow. We demonstrate that Lagrangian coherent structures (LCS), as identified by ridges in finite time Lyapunov exponents, are directly responsible for this coalescence of reactive scalar filaments. When highly concentrated filaments coalesce, reaction rates can be orders of magnitude greater than would be predicted in a well-mixed system. This is further supported by an idealized, analytical model that quantifies the competing effects of scalar dilution and coalescence. Chaotic flows, known for their ability to efficiently dilute scalars, therefore have the competing effect of organizing initially distant scalars along the LCS at timescales shorter than that required for dilution, resulting in reaction enhancement. This work is joint with K. Pratt and J. Crimaldi. Reaction Enhancement of Initially Distant Scalars by Lagrangian Coherent Structures | 13 November 2014 |

Guillaume Lajoie (Max Planck Institute for Dynamics and Self-Organization Göttingen) | **Abstract:** In this talk, I will discuss the use of Random Dynamical Systems (RDS) Theory as a framework to approach problems involving network dynamics in the brain. I will briefly outline the recent efforts to use RDS methods in the field of theoretical neuroscience before presenting recent results.
Large networks of sparsely coupled, excitatory and inhibitory cells occur throughout the brain. they support very complex computations, the exact mechanisms of which are poorly understood. For many models of these networks, a striking feature is that their dynamics are chaotic and thus, are sensitive to small perturbations. How does this chaos manifest in the neural code? Specifically, how variable are the spike patterns that such a network produces in response to an input signal?
To answer this, we derive a bound for a general measure of variability - spike-train entropy. The analysis is based on results from RDS theory and is complemented by detailed numerical simulations aimed at computing quantitative attributes of high-dimensional random strange attractors. This leads to important insights about the variability of multi-cell spike pattern distributions in large recurrent networks of spiking neurons responding to fluctuating inputs. Moreover, we show how spike pattern entropy is controlled by temporal features of the inputs. Our findings provide insight into how neural networks may encode stimuli in the presence of inherently chaotic dynamics. A Random Dynamical Systems framework to study encoding and variability in large neural networks | 13 November 2014 |

Alexander Gorodnik (University of Bristol) | **Abstract:** In this talk we discuss dynamics on flag manifolds, and in
particular, we will be interested in describing factors of such
dynamical systems. A well-known theorem of Margulis classifies
measurable factors, and an analogous result in the continuous category
has been established by Dani. We explain a classification of smooth
factors under some hyperbolicity assumptions. This is a joint work
with R. Spatzier. Dynamics on flag manifolds | 6 November 2014 |

Richard Sharp (University of Warwick) | **Abstract:** A beautiful theorem of Brooks says that, for a wide class of Riemannian manifolds, the bottom of the spectrum of the Laplacian on a regular cover is equal to the bottom of the spectrum of the base if and only if the covering group is amenable. In the case where the base manifold is a quotient of a simply connected manifold with pinched negative curvatures by a convex co-compact group, we will give a analogous results for critical exponents and for the growth of closed geodesics. This is joint work with Rhiannon Dougall. Critical exponents, growth and amenability | 30 October 2014 |

Laurent Stolovitch (University of Nice) | **Abstract:** We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the ``small divisors'' are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\al$-Gevrey vector field with an hyperbolic linear part admits a smooth $\be$-Gevrey transformation to a smooth $\be$-Gevrey normal form. The Gevrey order $\be$ depends on the rate of accumulation to $0$ of the small divisors. We show that a formally linearizable Gevrey smooth germ with the linear part satisfies Brjuno's small divisors condition can be linearized in the same Gevrey class. Smooth Gevrey normal forms of vector fields near a fixed point | 23 October 2014 |

Claude Baesens (University of Warwick) | **Abstract:** Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height E_c there is a
hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > E_c there are no bounded orbits. They called the bifurcation at E = E_c an abrupt bifurcation to chaotic scattering.
Our aim in this work is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems. Abrupt bifurcations in chaotic scattering: view from the anti-integrable limit | 16 October 2014 |

Ana Rodrigues (University of Exeter) | **Abstract:** In this talk, we will study the existence and non-existence of periodic orbits and limit cycles for planar polynomial differential systems of degree $n$ having $n$ real invariant straight lines taking into account their multiplicities. Periodic orbits for real planar polynomial vector fields of degree n having n invariant straight lines. | 9 October 2014 |

Gerhard Keller (University of Erlangen-Nürnberg) | **Abstract:** We study dynamical systems forced by a combination of random and deterministic
noise and provide criteria, in terms of Lyapunov exponents, for the existence of random
attractors with continuous structure in the bres. For this purpose, we provide suitable random versions of the semiuniform ergodic theorem and also introduce and discuss some basic concepts
of random topological dynamics. Random minimality and a theorem of Jaroslav Stark | 29 September 2014 |

Oleg Makarenkov (University of Texas at Dallas) | **Abstract:** Consider a couple of vector fields (F1,F2) and a couple of switching manifolds (S1,S2). Each trajectory x(t) follows the vector field F1 until x(t) crosses S1, where the system switches to the vector field F2 that governs the trajectory until it reaches S2 where the switch back to S1 occurs. The existence and stability of limit cycles in such a system is known since the classical work of Barbashin (1967) whose approach employs a Lyapunov-like technique. In this talk I show that the aforementioned cycles can be seen as a bifurcation from (0,0) when a suitably defined parameter crosses its bifurcation value. Bifurcation of limit cycles from a fold-fold singularity in switching systems | 16 September 2014 |

Jun-nosuke Teramae (Osaka University, Japan) | **Abstract:** Neurons in the brain sustain spontaneous ongoing activity with highly irregular spike trains. While the irregular activity has been regarded as ignorable background noise, recent experiments reveal that the spontaneous activity is significant player of computation in the brain. In this talk, introducing a recently proposed deterministic model of cortical network of spiking neurons that stably generates and maintains spontaneous ongoing activity with highly irregular spike trains, we study stochastic dynamics of the ongoing activity and discuss possible roles of the stochastic spiking for neural computation. Stochastic dynamics and computation in network models of cortical spiking neurons | 15 September 2014 |

Tiago Pereira (Imperial College London) | **Abstract:** Recent results reveal that typical real-world networks have various levels of connectivity. These networks exhibit emergent behaviour at various levels. Striking examples are found in the brain, where synchronisation between highly connected neurons coordinate and shape the network development. These phenomena remain a major challenge. I will discuss a probabilistic dimension reduction principle to describe the network dynamics. I show that, at large levels of connectivity, the high-dimensional network dynamics can be reduced to a few macroscopic equations. The strategy is to describe ensembles of random networks, and the dynamics almost every initial state. This reduction provides the opportunity to explore the coherent properties at various network connectivity scales. This is a joint work with Sebastian van Strien and Jeroen Lamb. Dynamics in Heterogeneous Networks: Emergence at Various Scales | 15 September 2014 |

Tsuyoshi Chawanya (Osaka University, Japan) | **Abstract:** Quasi-periodically forced logistic map system is one of the representative systems where strange non-chaotic attractors(SNAs) are observed. In this talk, we show that the numerical analysis on the bifurcation phenomena in QPLM can be carried out efficiently using a map describing the evolution of "line segment" in the logistic map system, and exhibit some of the obtained results including phase diagram with fine resolution indicating the existence of variety of bifurcation senario from a stable torus to a chaotic attractor with or without SNA in the middle. On bifurcation phenomena in the quasi-periodically driven logistic map system | 12 September 2014 |

Hiroshi Teramoto (Hokkaido University, Japan) | **Abstract:** Nonlinear resonance is one of the most efficient mechanisms for vibrational energy transfers among vibrational modes. In this talk, we propose a method to extract vibrational modes from a given time series to understand the underlying mechanism behind the energy transfers in terms of nonlinear resonances. Extracting coherent molecular vibrational modes by nonlinear time series analysis | 11 September 2014 |

Yuzuru Sato (Hokkaido University) | **Abstract:** Dynamics of dice roll on heterogeneous environments is studied in terms of random basin strucuture. Uncertainty exponents are numerically estimated in a random dynamical system and final state sensitivity under the presence of noise is investigated. Extracting random maps from experimental data of dice roll is also briefly discussed. Random basin in dice roll | 11 September 2014 |

Hiroki Sumi (Osaka University, Japan) | **Abstract:** In this talk, we consider random dynamical systems of complex polynomial maps on the Riemann sphere. It is well-known that for each rational map $f$ on the Riemann sphere with $\deg (f)\geq 2$, the Hausdorff dimension of the set of points $z$ in the Riemann sphere for which the Lyapunov exponent of the dynamics of $f$ is positive, is positive. However, we show that for generic i.i.d. random dynamical systems of complex polynomials, the following$B!!(Bholds. For all but countably many points $x$ in the Riemann sphere, for almost every sequence $\gamma =(\gamma _{1}, \gamma _{2}, \gamma _{3},\cdots )$ of polynomials, the Lyapunov exponent along $\gamma $ starting with $x$ is negative. Note that the above statement cannot hold in the usual iteration dynamics of a single rational map $f$ with $\deg (f)\geq 2.$ Therefore the picture of the random complex dynamics is completely different from that of the usual complex dynamics. We remark that even if the chaos of the random system disappears in the $C^{0}$ sense, the chaos of the system may remain in the $C^{1}$ sense, and we have to consider the ``gradation between chaos and order''. Negativity of Lyapunov exponents of generic random dynamical systems of complex polynomials | 9 September 2014 |

Masayuki Asaoka (Kyoto University, Japan) | **Abstract:** Theory of random dynamics studies properties of almost every
path of iterated function systems. In this talk, we discuss
how bad the dynamics can be for a special path and how does
it effect the average of dynamical quantity of iterated function
systems. More precisely, we prove the following theorem:
There exists an open subset of the set of smooth iterated
function systems on the unit interval with three generators
such that
(a) For all most path, the n-th iteration is uniform contraction,
and hence it has unique fixer point for any large n,
(b) For any generic element of U, the limsup of the average of
the number of fixed points of n-th iterations grows
superexpoentially.
The result illustrates that the behavior of the average in finite
time is different from that of a.e. paths.
We also proves that for generic element in U there exists a path
along which `any' dynamics is realized. Arbitrary growth of the number of periodic points and universal dynamics in one-dimensional semigroups | 9 September 2014 |

Christian Rodrigues (Max Planck Institute, Leipzig, Germany) | **Abstract:** Amongst the main concerns of Dynamics one wants to decide whether asymptotic states are robust under random perturbations. For practical applications, the randomly perturbed dynamics is given by a Markov chain model. Nevertheless, in order to derive conclusions from the mathematical perspective about stability, asymptotic states, invariance, etc., one is always enforced to use a model based on iteration of maps chosen at random with a given probability. The two approaches however are in general not equivalent. In this talk we systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. From this scheme, we not only deduce the representation by measurable and continuous random maps, but also obtain conditions for the to construct random diffeomorphisms from a given Markov chain. This is a joint work with Jost, and Kell. Stochastic stability and the representation of Markov chains by random maps | 8 September 2014 |

Rainer Klages (Queen Mary University of London) | **Abstract:** Dynamical systems having many coexisting attractors present interesting properties from both fundamental theoretical and modelling points of view. When such dynamics is under bounded random perturbations, the basins of attraction are no longer invariant and there is the possibility of transport among them. Here we introduce a basic theoretical setting which enables us to study this hopping process from the perspective of anomalous transport using the concept of a random dynamical system with holes. We apply it to a simple model by investigating the role of hyperbolicity for the transport among basins. We show numerically that our system exhibits non-Gaussian position distributions, power-law escape times, and subdiffusion. Our simulation results are reproduced consistently from stochastic Continuous Time Random Walk theory. Diffusion in randomly perturbed dissipative dynamics | 8 September 2014 |

Jennifer Creaser (University of Auckland) | **Abstract:** The well-known Lorenz system is classically studied via its reduction to
the one-dimensional Lorenz map, which captures the full behaviour of the
dynamics of the system. The reduction requires the existence of a stable
foliation. We study a parameter regime where this so-called foliation
condition fails and the Lorenz map no longer accurately represents the
dynamics. Hence, we study how the three-dimensional phase space is
organised by the global invariant manifolds of saddle equilibria and
saddle periodic orbits. We consider a previously unexplained phenomenon,
found by Sparrow in the 1980's where the one-dimensional stable manifolds
of the secondary equilibria flip from one side to the other. We
characterise this geometrically as a bifurcation in the alpha-limit of
these one-dimensional stable manifolds which we call an alpha-flip. We
find many such alpha-flips and, by following them in two parameters, show
that each ends at a different co-dimension two bifurcation point, known as
a T-point, many of which have not been found before. Moreover, we argue
that the alpha-flip is a precursor to the loss of the foliation condition. The Lorenz system near the loss of the foliation condition | 17 July 2014 |

Gary Froyland (University of New South Wales) | **Abstract:** Transfer operators provide a global description of dynamics and carry information that is di?fficult to extract by trajectory simulation. We describe theory and numerical methods for analysing (possibly time-dependent or random) chaotic dynamical systems, and applications in the natural and physical sciences. Transfer operators and dynamics | 14 July 2014 |

Nguyen Dinh Cong (Vietnam Academy of Science and Technology) | **Abstract:** Lyapunov exponents and Lyapunov spectrum are important tools in
the theory of dynamical systems. They characterize basic qualitative
properties of dynamical systems: stability, hyperbolicity, chaos, etc. In
this talk I will present some results on generic properties of Lyapunov
spectrum in the space of linear random dynamical systems. Generic properties of Lyapunov spectrum of
linear random dynamical systems | 14 July 2014 |

Sanjeeva Balasuriya (University of Adelaide) | **Abstract:** Within the context of fluid flows at scales ranging from the geophysical to the microfluidic, one might ask the complementary questions: (1) How does one define a flow barrier? and (2) Is it possible to quantify transport (i.e., a flux) across such barriers? In steady (autonomous) flows, these questions can be answered in terms of stable and unstable manifolds---whose intersection can only occur in restricted ways---across which there is zero flux. When the velocity field is unsteady (nonautonomous), such invariant manifolds move with time, and general intersections between them are possible. Defining these manifolds in realistic (unsteady, finite-time, discrete-valued) data sets is a challenge; accordingly, many heuristic definitions for flow barriers continue to be developed. Theoretical advances on nonautonomous stable and unstable manifolds are therefore of interest. Here, I address the question of obtaining these manifolds explicitly in a nonautonomously perturbed 2D situation, and in a nonautonomous nonchaotic 3D flow. The former requires the quantification of the tangential displacement of these manifolds under nonautonomous perturbations. The latter approach addresses a class of flows in which the time-variation of the manifolds can be explicitly established using exponential dichotomies. These models offer testbeds for the many proposed heuristics for identifying flow barriers in genuinely unsteady flows. Flow barriers and flux in unsteady flows | 2 July 2014 |

Jason Gallas (UFPB) | **Abstract:** Chaotic oscillations were recently discovered for some specific parameter
values in the de Pillis-Radunskaya model of cancer growth, a model including
interactions between tumor cells, healthy cells, and activated immune
system cells. I present high-resolution phase diagrams from a wide-ranging
systematic numerical classification of "all" dynamical states of the model
and their relative abundance. I characterize cell dynamics by two
independent and complementary types of stability diagrams: Lyapunov and
isospike diagrams.
The cancer model is shown to display stability phases regularly organized
in old and in many novel ways. In addition to spirals of stability,
the model displays very long sequences of zig-zag accumulations
and intertwined cascades of two- and three-chaos flanked stability islands
previously observed only in systems governed by delay-differential equations.
We also characterize a spike-adding mechanism underlying the systematic
complexification of regular wave patterns in generic flows when control
parameters are tuned continuously. Complex oscillations and chaos in a three-cell population model of cancer | 23 June 2014 |

James Robinson (University of Warwick) | **Abstract:** Suppose that we have a Lipschitz continuous differential equation on a Banach space $X$: $\dot x=f(x)$, where $\|f(x)-f(y)\|\le L\|x-y\|$. Using a geometric argument, Yorke showed that if $X$ is $R^n$ equipped with the usual Euclidean norm then any non-constant periodic orbit must have period at least $2\pi/L$. Busenberg, Fisher, and Martelli gave an analytical proof of the same result valid in any Hilbert space, and showed that in an arbitrary Banach space the minimal period is at least $6/L$. I will give proofs of these results, show that the minimal period is strictly larger than $6/L$ in any strictly convex Banach space (e.g. in $R^n$ with the $\ell^p$ norm, and discuss some related open problems. Minimal periods in Lipschitz differential equations | 19 June 2014 |

Dayal Strub (University Of Warwick) | **Abstract:** A transition state for a Hamiltonian system is a closed, invariant codimension-2 submanifold of an energy-level, spanned by two compact codimension-1 surfaces of unidirectional flux whose union locally separates the energy-level. This union, called a dividing surface, has locally minimal geometric flux through it and can therefore be used to find an upper bound on the rate of transport in Hamiltonian systems.
We shall first recall the basic transport scenario about an index-1 critical point of the Hamiltonian, and find transition states diffeomorphic to spheres for energies just above the critical one. This leads naturally to the question of what qualitative changes in the transition state may occur as the energy is increased further. We shall find that there is a class of systems for which the transition state changes diffeomorphism class via Morse bifurcations, and consider a number of examples.
This is joint work with Robert MacKay. Bifurcations of transition states | 12 June 2014 |

Alexandre Rodrigues (University of Porto) | **Abstract:** In this talk, we present a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes - we say that the nodes have different chirality. We show that in a $C^2$-open class of vector fields defined on a three-dimensional compact manifold, tangencies of the invariant manifolds of two hyperbolic saddle-foci occur at a full Lebesgue measure set of the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. Dense heteroclinic tangencies near a Bykov cycle | 10 June 2014 |

Alexandre Rodrigues (University of Porto) | **Abstract:** In this talk, we characterise the set of $C^1$ area-preserving maps on a surface displaying a reversing isometry $R$ of degree 2 (involution). We show that $C1$-generic $R$-reversible area-preserving maps are Anosov or else Lebesgue almost every orbit displays zero Lyapunov exponents. This result generalizes Bochi-Mañé Theorem for the class of reversing-symmetric maps. Generic Area preserving reversible diffeomorphisms | 9 June 2014 |

Vassilis Rothos (Aristotle University of Thessaloniki) | Bifurcation of Travelling Waves in Implicit Nonlinear Lattices: Applications in magnetic metamaterials | 4 June 2014 |

Claudia Wulff (University of Surrey) | **Abstract:** Relative equilibria and relative periodic orbits (RPOs)
are ubiquitous in symmetric Hamiltonian systems
and occur for example in celestial mechanics, molecular dynamics and rigid body
motion. Relative equilibria are equilibria and RPOs
are periodic orbits of the symmetry
reduced system.
Relative Lyapunov centre bifurcations are bifurcations of relative periodic orbits
from relative equilibria corresponding to Lyapunov centre bifurcations of the symmetry
reduced dynamics.
In this talk we prove a relative Lyapunov centre theorem by combining
recent results on persistence of RPOs in Hamiltonian systems with a symmetric
Lyapunov centre theorem of Montaldi et al.
We then develop numerical methods for the detection of
relative Lyapunov centre bifurcations along branches of RPOs and for their computation.
We apply our methods to Lagrangian relative equilibria of the N-body problem. Relative Lyapunov centre bifurcation | 29 May 2014 |

Jason Gallas (UFPB) | **Abstract:** A central problem in modern cryptography is the factorization of large
integers involved in theoretically breakable but computationally secure
mechanisms used to protect data. I will discuss a method to solve an
analogous but more general problem for functions, not numbers:
the factorization of exceptionally large polynomials defining the orbital
points of periodic orbits of the quadratic (logistic) map, a paradigmatic
discrete-time dynamical system of algebraic origin. The method is based
on an infinite set of nonlinear transformations whose zeros are all
preperiodic points of the dynamics. Forward iteration from such preperiodic
points opens access to orbits with arbitrarily long periods which cannot
be reached using standard multivalued inverse functions.
As a concrete example, I show how to extract expeditiously a long periodic
orbit buried in a polynomial with degree larger than a billion.
Even rough numerical approximations of the zeros suffice to obtain
long orbits precisely. Factorization of exceptionally long periodic orbits of the quadratic map | 22 May 2014 |

Daniel Karrasch (ETH Zurich) | **Abstract:** In this talk I give an overview on the geodesic theory of Lagrangian Coherent Structures (LCS) in two-dimensional, finite-time nonautonomous dynamical systems. LCS are exceptional material lines that act as cores of observed tracer patterns in finite-time (fluid) flows. The talk is organized in two parts. In the first part, I briefly present the variational derivation of hyperbolic LCS, as due to Farazmand, Blazevski & Haller. I then describe an attraction-based approach to hyperbolic LCS (joint work with M. Farazmand and G. Haller), which correponds to a paradigm shift in hyperbolic LCS theory. In the second part, I briefly present the variational derivation of elliptic LCS, which are buidling blocks of coherent Lagrangian vortices, as due to Haller & Beron-Vera. I then present an automated detection method based on index theory for line fields (joint work with F. Huhn and G. Haller). Overview on geodesic Lagrangian Coherent Structures | 22 May 2014 |

Kathrin Padberg-Gehle (TU Dresden) | **Abstract:** Numerical methods involving transfer operators have only recently been recognized as powerful tools for analyzing and quantifying transport and mixing in time-dependent systems. This talk discusses several different constructions that allow us to extract coherent structures and dynamic transport barriers in nonautonomous dynamical systems. Apart from exploiting spectral properties of the transfer operator, we pinpoint transport barriers as regions of low predictability. Predictability can be very efficiently measured via the growth of entropy that is experienced by a small localised density under the evolution of the numerical transfer operator. Transport, mixing and predictability | 15 May 2014 |

Julia Slipantschuk (Queen Mary University) | **Abstract:** Spectral data of transfer operators yield insight into fine
statistical properties of the underlying dynamical system, such as rates of mixing.
In this talk, I will describe the spectral structure of transfer operators
associated to analytic expanding circle maps. For this, I will first
derive a natural representation of the respective adjoint operators. For expanding
circle maps arising from finite Blaschke products, this representation takes
a particularly convenient form, allowing to deduce the entire spectra
of the corresponding transfer operators. These spectra are completely
determined by the multipliers of attracting fixed points of the Blaschke products. Spectral structure of transfer operators for expanding circle maps | 8 May 2014 |

Heather Reeve-Black (Queen Mary University) | **Abstract:** We consider a one-parameter family of invertible maps of a two-dimensional lattice,
obtained by applying round-off to planar rotations.
We let the angle of rotation approach $\pi/2$, and show that the system exhibits a
failure of shadowing: the limit of vanishing discretisation corresponds to a piecewise-affine Hamiltonian flow,
whereby the plane foliates into invariant polygons with an increasing number of sides.
Considered as perturbations of the piecewise-affine flow, the lattice maps assume
a different character, described in terms of strip maps, a variant
of those found in outer billiards of polygons. Furthermore the flow is nonlinear
(unlike the rotation) and a suitably chosen Poincare return map is a twist map.
We show that the motion at infinity, where the invariant polygons approach circles,
is a dichotomy: there is one regime in which the nonlinearity tends to zero, leaving only the perturbation,
and a second where the nonlinearity dominates.
In the domains where the nonlinearity remains, numerical evidence
suggests that the distribution of the periods of orbits is consistent with that of random dynamics,
whereas in the absence of nonlinearity, the fluctuations result in intricate discrete resonant structures. Near-Integrability in a Family of Discretised Rotations | 1 May 2014 |

Boumediene Hamzi (Yildiz Technical University) | **Abstract:** We introduce a data-based approach to estimating key
quantities which arise in the study of nonlinear control and random
dynamical systems. Our approach hinges on the observation that much of
the existing linear theory may be readily extended to nonlinear
systems - with a reasonable expectation of success - once the
nonlinear system has been mapped into a high or infinite dimensional
Reproducing Kernel Hilbert Space. In particular, we develop
computable, non-parametric estimators approximating controllability
and observability energy functions for nonlinear systems, and study
the ellipsoids they induce. It is then shown that the controllability
energy estimator provides a key means for approximating the invariant
measure of an ergodic, stochastically forced nonlinear system. We also
apply this approach to the problem of model reduction of nonlinear
control systems. In all cases the relevant quantities are estimated
from simulated or observed data. These results collectively argue that
there is a reasonable passage from linear dynamical systems theory to
a data-based nonlinear dynamical systems theory through reproducing
kernel Hilbert spaces.
This is joint work with J. Bouvrie (MIT). On Control and Random Dynamical Systems in Reproducing Kernel
Hilbert Spaces | 24 April 2014 |

Sebastian Hage-Packhäuser (University of Paderborn) | **Abstract:** Numerous dynamical systems describing real world phenomena exhibit a characteristic fine structure which stems from the interaction of many dynamic instances. Furthermore, since reality crucially depends on time, such dynamical system networks are generally subject to temporal changes. Particularly in applications involving technology, this temporal evolution often occurs as a consequence of instantaneously time-varying network structures.
In this talk, time-varying networks of dynamical systems are discussed in terms of hybrid dynamical systems with a special consideration of symmetries which are naturally due to the network structures involved. By means of the recent notion of hybrid symmetries, a hybrid symmetry framework is presented and symmetry-induced switching strategies are investigated - from a structural point of view, but also with regard to stability.
Switching Dynamical System Networks in the Light of Hybrid Symmetry | 2 April 2014 |

Eduardo Garibaldi (UNICAMP) | **Abstract:** In this talk, we discuss the existence of minimizing configurations
associated with generalized Frenkel-Kontorova models on quasi-crystals.
This is a joint work with Samuel Petite (Université de Picardie) and
Philippe Thieullen (Université de Bordeaux). The Frenkel-Kontorova model for almost-periodic environments | 31 March 2014 |

Frits Veerman (University of Oxford) | **Abstract:** The study of localised structures in systems of reaction-diffusion equations is a very active field of research. The nonlinear nature of the equations often makes it difficult to go beyond standard analysis, i.e. Turing patterns. The presence of small parameters, in particular the diffusion ratio, can be used to successfully exploit these nonlinearities and to invoke geometric singular perturbation theory to prove, by construction, the existence of localised patterns. Even the stability of these patterns can be addressed: using Evans function techniques, the spectrum of the pulse can be determined to leading order. This method, which can be applied to a very general class of reaction-diffusion systems, will be applied to a example system, where previously unobserved behaviour is found and analysed.
This is joint work with A. Doelman, University of Leiden. Pulses in singularly perturbed reaction-diffusion systems | 28 March 2014 |

Evamaria Ruß (Alpen Adria University Klagenfurt) | **Abstract:** The dichotomy spectrum (also known as Sacker-Sell or dynamical spectrum) is a crucial spectral notion in the theory of dynamical systems. In this talk we study the dichotomy spectrum for linear difference equations with an infinite-dimensional state space. In general we cannot expect a nice structure of the dichotomy spectrum like in the finite dimensional case, but compactness properties of the transition operator provide a more regular spectrum. Finally, we have a look at various evolutionary differential equations in order to illustrate possible applications. Dichotomy Spectrum in Infinite Dimensions | 27 March 2014 |

Ian Morris (University of Surrey) | **Abstract:** The binary Euclidean algorithm is a variation on the classical Euclidean algorithm which is designed to take advantage of the efficiency of division by two on a binary computer. Whereas the classical Euclidean algorithm can be understood in terms of the dynamics of the continued fraction transformation on the unit interval, the analysis of the binary Euclidean algorithm requires the use of a random dynamical system. I will describe some of my recent work on the analysis of this algorithm via the Ruelle transfer operator associated to this random dynamical system and its connection with a conjecture of D. E. Knuth.
A random dynamical model for the binary Euclidean algorithm | 20 March 2014 |

Masayuki Asaoka (Kyoto University) | **Abstract:** In 1999, Kaloshin showed that persistence of homoclinic tangency
implies local genericity of arbitrary fast growth of the number
of periodic orbits. Local genericity of universal dynamics
(Bonatti-Diaz, Turaev) also implies local genericity of such
pathological behavior. But, up to now, all known mechanism of
abnormal growth is based on homoclinic tangency.
A natural question is whether generic partially hyperbolic
system can exhibit arbitrary fast growth of the number of
periodic orbit or not. As a step to answer this problem, we
give a solution to a corresponding problem for 1-dimensional
iterated function systems:
Theorem: There exists an open set of C^r 1-dimensional
iterated function system in which generic system exhibits
arbitrary fast growth of the number of periodic points.
If possible, we also discuss a work in progress which extends
the above theorem to 3-dimensional partially hyperbolic
systems.
This is a joint work with K.Shinohara and D.Turaev. Arbitrary fast growth of the number of periodic orbits in
1-dimensional iterated function systems | 14 March 2014 |

Piotr Slowinski (University of Warwick) | **Abstract:** In this talk I will present bifurcation analysis of a semiconductor laser receiving delayed filtered optical feedback from two filter loops (2FOF). Specifically, we compute the basic cw-solutions (called external filtered modes EFM) of an underlying delay differential equation model. The EFMs as represented as surfaces in the space of filter phase difference versus frequency and inversion of the laser. In this way, I am able to present a comprehensive geometric picture of how the EFM structure and stability depends on parameters, including the filter detunings and delays.
Overall, the study of the EFM-surface is a geometric tool for the multi-parameter analysis of the 2FOF laser, which provides comprehensive insight into the solution structure and dynamics of the system. Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops | 13 March 2014 |

Sofia Castro (University of Porto) | **Abstract:** It is well-known that some cycles in heteroclinic networks appear more frequently in simulations than others. Making use of a convenient notion of stability and of the stability index introduced by Podvigina and Ashwin [3], I shall report on the possible relative stability of two cycles of type B inside a network with no other cycles. This is recent work with Alexander Lohse [2].
Another illustrative example of the importance of relative stability of cycles in networks is provided by the quotient network arising in the Rock-Scissors-Paper. This consists of older work with Manuela Aguiar [1] and work in progress with Yuzuru Sato.
[1] M.A.D. Aguiar and S.B.S.D. Castro, Chaotic switching in a two-person
game, Physica D: Nonlinear Phenomena, vol. 239 (16), 1598-1609 (2010)
[2] S.B.S.D. Castro and A. Lohse, Stability in simple heteroclinic networks in R^4, arXiv:1401.3993 [math.DS] (2014)
[3] O. Podvigina and P. Ashwin, On local attraction properties and a stability index for heteroclinic connections,
Nonlinearity, Vol. 24, 887-929 (2011) Stability of cycles in heteroclinic networks | 13 March 2014 |

Robert Szalai (University of Bristol) | **Abstract:** Friction and impact are two non-smotth nonlinearities that are of great interests to engineers. The problem with non-smooth phenomena is that they introduce a great deal of uncertainty of into model predictions. Specifically it is possible to lose forward time uniqueness and have generic models that are structurally unstable. In this talk I will introduce a modelling framework that hopefully resolves these problems. First I demonstrate how non-unique solutions arise in mechanical systems composed of rigid bodies. It turns out that the major component of such singular behaviour is the rigid body assumption. In reality nothing is ideally rigid. To resolve such singularities the elasticity needs to be taken into account. These elastic systems are described by partial differential equations (PDE), which make the analysis complicated. Hence to resolve the problem I introduce a model reduction technique that turns those PDEs into low dimensional delay equations. It turns out that the only condition for regular dynamics is to have a finite wave speed within the mechanism. The method is general and does not rely on the existence of travelling wave solutions. How to resolve singularities in non-smooth systems using better
physical models | 6 March 2014 |

Hans-Henrik Rugh (University of Paris-Sud) | **Abstract:** (Work in progress with O. Bandtlow).
We consider uniformly expanding interval maps that are Markov.
Introducing a finite number of holes, restricting the map to the
remaining set. We show that the entropy of the restricted map
is Hölder continuous with respect to the hole position and size
under a non-degeneracy condition. The Hölder exponent
is related to the entropy itself and expansion rates of the map.
Examples and counter-examples, with numerics illustrates
the result. On the Hölder continuity of the entropy for expanding Interval maps with holes | 6 March 2014 |

Stefano Marmi (Scuola Normale Superiore, Pisa) | **Abstract:** Addressing a question raised by Kolmogorov and Herman, we show that KAM curves of area-preserving twist maps are uniquely determined by their knowledge on a set of positive 1-dimensional Hausdorff measure in frequency space. This result is obtained by complexifying the rotation number and by an extension of the classical theory of quasianalytic functions. The parametrization of KAM curves is naturally defined in a complex domain containing the real Diophantine frequencies and real frequencies constitute a natural boundary for the analytic continuation from the Weierstrass point of view because of the density of the resonances. Natural boundaries and uniqueness of KAM curves | 6 March 2014 |

Jimmy Tseng (University of Bristol) | **Abstract:** We show that, for two commuting automorphisms of the d-torus, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity and the Ledrappier-Young formula, we show that the set of points that have dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. This mixed case, dense orbit under one map and nondense orbit under the other, is much more delicate than the other two possible cases.
Our technique can also be applied to other settings. For example, we show the analogous result for two elements of the Cartan action on compact higher rank homogeneous spaces.
This is joint work with V. Bergelson and M. Einsiedler. Simultaneous dense and nondense orbits for commuting maps | 27 February 2014 |

Ilies Zidane (Paul Sabatier University) | **Abstract:** Yoccoz gave a sufficient arithmetical condition of linearization of fixed points of holomorphic germs with
multiplier exp(2 i \pi a) where a is an irrational number: f(z)=exp(2 i \pi a) z+O(z^2). He also proved that this condition
is optimal for quadratic polynomials. We will discuss this optimality for cubic polynomials and quadratic rational maps.
We will see how is it related to the size of Siegel disks and parabolic implosion/renormalization. This leads to the
study of slices of bifurcation locus where some surprising, unexpected and complicated phenomenons occur due to
the interaction between the two critical points. We also investigate some virtual slices arising as geometric limits
(parabolic enrichment) of dynamical systems. We seek analogues of Zakeri curves (the locus where the two critical
points lie at the boundary of the Siegel disk) in these slices, when the rotation number is not of bounded type, and
even, for Cremer slices. Given a Siegel slice, the logarithm of the conformal radius of the Siegel disk is a subharmonic
function, whose Laplacian is therefore a measure which gives a new viewpoint as well as a lot of information. On the bifurcation locus of cubic polynomials and the size of Siegel disks | 25 February 2014 |

Oleg Kozlovski (University of Warwick) | **Abstract:** In this talk we will discuss if maps in a "typical" family of
maps of an interval or a circle have bounded number of attracting
trajectories. The answer is not straightforward and
depends on what "typical" means and on the smoothness of the maps. Hilbert-Arnold problem for 1D maps | 13 February 2014 |

David Lloyd (University of Surrey) | **Abstract:** Experiments involving a plate of magnetic fluid in the presence of a magnetic field have shown that it is possible for stable localised spots on the surface of the magnetic to exist. Building on work analysing localised patterns in reaction-diffusion equations, we try to explain the nucleation of such spots for the magnetic fluid and some of their properties away from onset. In particular, I will show that the equations governing the free-surface of the magnetic fluid obey an energy principle. Furthermore, an approximate Hamiltonian formulation can be found near onset that simplifies a centre-manifold reduction for patterns in the one-dimensional case. Finally, we show some numerical continuation results for 2D and 3D localised Ferropatterns and relate the results to work on reaction-diffusion systems in the presence of a conserved quantity. Nucleation of localised Ferrosolitons and Ferro-patterns | 6 February 2014 |

Luciana Luna Anna Lomonaco (Roskilde University) | **Abstract:** A polynomial-like mapping is a proper holomorphic map f : U' -> U, where
U' and U isomorphic to a a disk, and U' compactly contained in U. This definition captures the behaviour of a
polynomial in a neighbourhood of its filled Julia set. A polynomial-like map of
degree d is determined up to holomorphic conjugacy by its internal and external
classes, that is, the (conjugacy classes of) the restrictions to the filled
Julia set and its complement. In particular the external class is a degree
d real-analytic orientation preserving and strictly expanding self-covering of
the unit circle: the expansivity of such a circle map implies that all the
periodic points are repelling, and in particular not parabolic.
We extended the polynomial-like theory to a class of parabolic mappings
which we called parabolic-like mappings. A parabolic-like mapping is an object
similar to a polynomial-like mapping, but with a parabolic external class;
that is to say, the external map has a parabolic fixed point, whence the
domain is not contained in the codomain.
In this talk we present the parabolic-like mapping theory. We give a sketch of the proof of the Straightening Theorem
for parabolic-like mappings, which states
that every degree 2 parabolic-like mapping is hybrid equivalent to a member of the family of quadratic rational maps of the
form P_A(z)=z+ 1/z+ A, where A is a complex number.
Then we will consider families of parabolic-like mappings, state the main result in this setting and give an application. Parabolic-like mappings | 31 January 2014 |

Nick Sharples (Imperial College London) | **Abstract:** In this talk I will outline the renormalization theory for irregular Ordinary Differential Equations, introduced by DiPerna & Lions, which provides existence and uniqueness of generalised flow solutions for vector fields that have limited (e.g. Sobolev) regularity. I will discuss a recent extension of these results to include vector fields with a set of 'bad' singularities, where the vector field is not locally of bounded variation, provided that this singular set is sufficiently small in a fractal sense (joint with E. Miot).
I will consider the non-autonomous case where the singular set is a subset of the extended phase space. In this setting the appropriate notion of fractal dimension is that of an anisotropic 'codimension print' in which the spatial and temporal detail of the singular set can be distinguished. I will relate this esoteric notion of dimension to the more familiar box-counting dimension (joint with J.C. Robinson) providing straightforward criteria for the existence and uniqueness of non-autonomous generalised flows. Irregular ODEs: renormalization and geometry | 30 January 2014 |

Konstantin Khanin (Loughborough University) | **Abstract:** In this talk we shall report on recent progress in rigidity theory for nonlinear interval exchange transformations corresponding to cyclic permutations. Such maps can be viewed as circle homeomorphisms with multiple break points. We shall discuss both recent results on renormalizations of such maps in case of one break point (joint with S. Kocic and A. Teplinsky), and extension to the multiple-break setting (based on work in progress with A. Teplinsky). On rigidity for cyclic nonlinear interval exchange transformations | 23 January 2014 |

Anna Cherubini (University of Salento) | **Abstract:** This study deals with the identification of early-warning signals for desertification in fragile ecosystems such as arid or semi-arid ones. Literature on this topic shows that vegetation patchiness in semi-arid ecosystems can lead to the identification of indicators for an approaching transition to desertification. In particular, the analysis of the spatial fluctuations of vegetation patterns suggests that a significant deviation from power laws in vegetation patch size distributions is a reliable signal for an approaching desertification. In this work we analysed a model for semi-arid ecosystems based on a stochastic cellular automaton (CA) depending on a number of parameters (accounting for external stresses, soil quality, water, interactions between plants). We investigated the time fluctuations properties of the quantities associated with the steady states of the CA and we found that other possible and earlier signals are related to percolation thresholds and to the time fluctuations distribution of the biggest cluster size. Time fluctuations of vegetation patterns and early warnings for desertification | 21 January 2014 |

Raffaele Vitolo (University of Salento) | **Abstract:** Homogeneous differential-geometric Poisson brackets were
introduced by Dubrovin and Novikov in 1984. Such operators appear in
many integrable systems. First order operators have been extensively
studied so far. In this talk we will devote ourselves to the
classification of third order operators with non-degenerate leading
term. Starting from old results by one of us (GVP) we prove that
such operators are completely characterized by their leading term, which is a Monge metric. Monge metrics are distinguished pseudo-Riemannian metrics which are in bijection with quadratic line complexes.
Quadratic line complexes are algebraic varieties for which classification results are known in the 2 and 3 component cases.
Using such results we are able to provide a classification of Hamiltonian operators in 2 and 3 components, together with some hints on how to solve the problem in a higher number of components.
Joint work with E.V. Ferapontov, M.V. Pavlov, G.V. Potemin. On the classification of homogeneous differential-geometric
third-order Poisson brackets | 16 January 2014 |

Rainer Klages (Queen Mary University) | **Abstract:** My talk is about the impact of spatial [1] and temporal [2] random perturbations on
diffusion in chaotic dynamical systems. As an example, I consider deterministic random
walks on a one-dimensional lattice. The system is modeled by a piecewise linear map defined
on the unit interval which depends on two control parameters and is lifted onto the whole
real line. Computer simulations show a rich scenario in the diffusion coefficient of this model
by increasing the perturbation strength. Typical signatures of the transition from small to
large perturbations are multiple suppression and enhancement of diffusion by approaching
basic asymptotic laws for large perturbation strength. These results are reproduced by
simple approximations based on the parameter dependence of the unperturbed deterministic
diffusion coefficient [3].
[1] R.Klages, Phys. Rev. E 65, 055203(R) (2002)
[2] R.Klages, Europhys. Lett. 57, 796 (2002)
[3] R.Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical
Mechanics, Advanced Series in Nonlinear Dynamics Vol.24 (World Scientific, Singapore,
2007), Part 1. Chaotic diffusion in randomly perturbed dynamical systems | 12 December 2013 |

Sigurður Hafstein (Reykjavik University) | **Abstract:** Lyapunov functions can characterize many fundamental properties of dynamical systems as attractor-repeller pairs, basins of attraction, and the chain-recurrent set. We discuss an algorithm to compute continuous and piecewise affine (CPA) Lyapunov functions for continuous nonlinear systems by linear optimization. The algorithm can be adapted to compute Lyapunov functions for continuous differential inclusions and discrete systems. We further discuss how the algorithm can be combined with faster methods with less concrete bounds, e.g. the radial-basis-functions collocation method, to deliver true Lyapunov functions
comparatively fast. Computation of CPA Lyapunov functions | 9 December 2013 |

Sebastian Wieczorek (University of Exeter) | **Abstract:** Rate-induced bifurcations occur in forced systems where there is a stable state for every fixed level of forcing. When forcing varies in time, the position of the stable state changes and the system tries to keep pace with the changes. However, some systems fail to adiabatically follow the continuously changing stable state and destabilise above some critical rate of forcing.
Scientists often find rate-induced bifurcations counter-intuitive because there is no obvious loss of stability and no obvious threshold. On the other hand, these non-autonomous instabilities cannot in general be described by classical bifurcations or asymptotic approaches and remain fairly unexplored. I will present an approach based on geometrical singular perturbation theory to study critical rates of change and non-obvious thresholds. I will also discuss repercussions for climate change policy making which currently focuses on critical levels of the atmospheric temperature whereas the critical factor may be the rate of warming rather than the temperature itself. Rate-induced bifurcations: critical rates, non-obvious thresholds and failure to adapt to changing external conditions | 5 December 2013 |

Jürgen Knobloch (TU Ilmenau) | **Abstract:** Homoclinic snaking refers to the continuation curves of homoclinic
orbits near a heteroclinic cycle, which connects an equilibrium and a
periodic orbit in a reversible Hamiltonian system. We consider
non-reversible perturbations of this situation and show analytically
that such perturbations typically lead to either infinitely many closed
continuation curves (isolas) or to two snaking continuation curves,
which follow the primary sinusoidal continuation curves alternately
(criss-cross snaking). Non-reversible perturbations of homoclinic snaking scenarios | 28 November 2013 |

Evelyne Miot (CNRS, Paris) | **Abstract:** A system of equations combining the 1D Schrödinger equation and the point vortex system has been derived by Klein, Majda and Damodaran to modelize the evolution of nearly parallel vortex filaments in 3D incompressible fluids. In this talk I will describe some dynamics for this system such as travelling waves, collisions and finite-time blow-up. I will finally present the case of pairs of filaments. This is joint work with Valeria Banica and Erwan Faou. Some examples of dynamics for nearly parallel vortex filaments | 21 November 2013 |

Janosch Rieger (Goethe University Frankfurt) | **Abstract:** The theory of finite element methods is one of the success stories of modern
mathematics. It is therefore reasonable to ask what set-valued numerical analysis can
learn from this field. The main aim of this talk is to popularise the concept of set
spaces that are defined in terms of a generalization of the well-known support
function. This concept seems to be the language in which the central ideas behind finite
element methods can be transferred to the computation of sets.
After a formal introduction to set spaces, some graphic examples of such spaces will
be given, and the relationship between conventional representations of sets and set
spaces will be explained. Some preliminary results will be shown and commented. Spaces of non-convex sets and their potential for set-valued numerical analysis | 14 November 2013 |

Ale Jan Homburg (University of Amsterdam) | **Abstract:** I'll consider iterated function systems generated by finitely many diffeomorphisms on compact manifolds. I'll discuss aspects of their dynamics, in particular minimality and synchronization.
These iterated function systems play a pivotal role in the study of dynamical systems: they correspond to dynamical systems of skew product type and provide examples of "partially hyperbolic dynamical systems". I'll discuss how iterated function systems are giving new results and insights in the study of partially hyperbolic dynamics. Iterated function systems, skew product dynamics, partially hyperbolic dynamics | 7 November 2013 |

Gioia Vago (Université de Bourgogne) | **Abstract:** The Ogasa invariant is defined for any compact manifold in
any dimension. Roughly speaking, it is computed on the largest regular
level of a slimmest Morse function.
The minimax procedure underlying its definition makes its exact computation extremely hard, not to say impossible, in the very general
case. However, it is crucial to know as much as we can about its
behaviour, because this invariant contains precious and sharp information about how much structure the manifold has, and therefore it can be used as a detector of singularities.
In dimension 2, the computation of this invariant is straightforward.
Dimension 3 is the first non-trivial case. As a result of a joint work with Michel Boileau (Univ. Aix-Marseille, France), now we have a global qualitative and quantitative understanding of what this dynamical invariant measures, and how it is related to other topological and algebraic invariants of the underlying manifold. The Ogasa invariant in dimension 3 | 31 October 2013 |

Jan Sieber (University of Exeter) | **Abstract:** If one wants to perform bifurcation analysis for a given smooth low-dimensional dynamical system of interest, one can use a range of ready-made numerical tools such as AUTO, MatCont or CoCo. I will show which aspects of this analysis can be carried out also in physical experiments. One typical limitation in physical experiments is that one cannot set the internal state of the system at will. An example, tracking branches of unstable periodic orbits around a saddle-node, in a simple mechanical experiment will demonstrate the basic principle.
Other potential applications are:
* finding the flow on unstable parts of the slow manifold in slow-fast systems (the unstable parts of so-called canards),
* study of the collapse of long chaotic transients in high-dimensional systems.
[joint work with David Barton (Bristol), Oleh Omelchenko, Matthias Wolfrum (WIAS Berlin, Germany)] Bifurcation Analysis for Experiments | 17 October 2013 |

Dmitry Turaev (Imperial College London) | **Abstract:** Let a real-analytic Hamiltonian system have a normally-hyperbolic cylinder such that the Poincare map on the cylinder has a twist property. Let the stable and unstable manifolds of the cylinder intersect transversely in a certain strong sense. The homoclinic channel is a small neighbourhood of the union of the cylinder and the homoclinic. We show that generically (in the real-analytic category) in the channel there always exist orbits which move from one end of the cylinder to the other. This opens a way of showing that given an integrable system with 3 or more degrees of freedom, for arbitrarily small generic Hamiltonian perturbations the change of action variables along resonances is bounded away from zero. Arnold diffusion in a priori chaotic systems | 10 October 2013 |

Davoud Cheraghi (Imperial College London) | **Abstract:** The local and global dynamics of holomorphic maps near fixed points with asymptotic irrational rotation has been extensively studied through various methods over the last decades. The problem involves delicate interplay between the Diophantine approximation of the irrational rotation, "small divisors", and the distortion properties of holomorphic maps. In this talk we report on historical developments in the subject and discuss some recent breakthrough using renormalization techniques. Small divisors in holomorphic dynamics | 8 October 2013 |

Erik Bollt (Clarkson University) | **Abstract:** Mixing, and coherence are fundamental issues at the heart of understanding fluid dynamics and other non- autonomous dynamical systems. Only recently has the notion of coherence come to a more rigorous footing, and particularly within the recent advances of finite-time studies of nonautonomous dynamical systems.. Here we define shape coherent sets which we relate to measure of coherence in differentiable dynamical systems from which we will show that tangency of finite time stable foliations (related to a forward time perspective) and finite time unstable foliations (related to a “backwards time" perspective) serve a central role. This perspective is agreeable with the recent theory of geodesics by Haller et. al. derived from a variational principle of geodesics. We develop zero-angle curvers, meaning non-hyperbolic splitting, by continuation methods in terms of the implicit function theorem, from which follows a simple ODE description of the boundaries of shape coherent sets. Differential Geometry Perspective of Shape Coherence and Curvature Evolution by Finite-Time Nonhyperbolic Splitting
| 3 October 2013 |

Sebastian van Strien (Imperial College London) | **Abstract:** These lectures will review some results about the dynamics of interval maps, focusing especially on recent results which rely on tools from complex analysis. Topics covered will include: topological and topological properties of the Julia sets, density of maps with good behaviour (hyperbolic maps) There are no specific prerequisites for this course, apart from a basic course in complex analysis.
A related, and more extensive MSc course, will be taught jointly with Davoud Cheraghi in the 2nd term (TCC). Holomorphic dynamics of real interval maps: chaos and hyperbolicity (Part III) | 3 October 2013 |

Sebastian van Strien (Imperial College London) | **Abstract:** These lectures will review some results about the dynamics of interval maps, focusing especially on recent results which rely on tools from complex analysis. Topics covered will include: topological and topological properties of the Julia sets, density of maps with good behaviour (hyperbolic maps) There are no specific prerequisites for this course, apart from a basic course in complex analysis.
A related, and more extensive MSc course, will be taught jointly with Davoud Cheraghi in the 2nd term (TCC).
Holomorphic dynamics of real interval maps: chaos and hyperbolicity (Part II) | 2 October 2013 |

Sebastian van Strien (Imperial College London) | **Abstract:** These lectures will review some results about the dynamics of interval maps, focusing especially on recent results which rely on tools from complex analysis. Topics covered will include: topological and topological properties of the Julia sets, density of maps with good behaviour (hyperbolic maps) There are no specific prerequisites for this course, apart from a basic course in complex analysis.
A related, and more extensive MSc course, will be taught jointly with Davoud Cheraghi in the 2nd term (TCC).
Holomorphic dynamics of real interval maps: chaos and hyperbolicity (Part I) | 1 October 2013 |

Bob Rink ( Vrije Universiteit Amsterdam
) | **Abstract:** A classical result of Aubry and Mather states that Hamiltonian twist maps have orbits of all rotation numbers. Analogously, one can show that certain ferromagnetic crystal models admit ground states of every possible mean lattice spacing. In this talk, I will show that these ground states generically form Cantor sets, if their mean lattice spacing is an irrational number that is easy to approximate by rational numbers. This is joint work with Blaz Mramor.
Ferromagnetic crystals and the destruction of minimal foliations | 12 July 2013 |

Bob Rink (Vrije Universiteit Amsterdam) | **Abstract:** Dynamical systems with a coupled cell network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web.
A network structure can have a strong impact on the behaviour of a dynamical system. For example, it has been observed that networks can robustly exhibit (partial) synchronisation, multiple eigenvalues and degenerate bifurcations. In this talk I will explain how semigroups and their representations can be used to understand and predict these phenomena. As an application of our theory, I will discuss how a simple feed-forward motif can act as an amplifier.
This is joint work with Jan Sanders. Using semigroups to study coupled cell networks | 10 July 2013 |

June Barrow-Green (The Open University) | **Abstract:** In October 1912, the young American mathematician G. D. Birkhoff 'astonished the mathematical world' by providing a proof of Poincaré's last geometric theorem. The theorem, which was connected to Poincaré's long standing interest in the periodic solutions of the three-body problem, had been proposed by him only months before he died. Birkhoff continued to work on aspects of dynamical systems throughout his career, his aim being to create a general theory. Many of his ideas are contained in his book Dynamical Systems (1927), the first book to develop the qualitative theory of systems defined by differential equations and where he effectively 'created a new branch of mathematics separate from its roots in celestial mechanics and making broad use of topology'. G. D. Birkhoff and the development of dynamical systems theory
| 26 June 2013 |

Ken Palmer (Providence University, Taiwan) | **Abstract:** Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property,
that is, any pseudo (or approximate) orbit has
a true orbit nearby. In fact, the system has the
Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk
this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin
and S. Tikhomirov, will be described.
Numerical aspects: This is joint work with Brian Coomes and H\" useyin Ko\c cak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented.
Given a suitable approximate connecting orbit and assuming
that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently
``good''.
It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A
numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).
Some theoretical and numerical aspects of shadowing | 26 June 2013 |

Pablo Guarino (Universidade de Sao Paulo) | **Abstract:** The so-called critical circle maps are orientation-preserving smooth circle homeomorphisms having critical points (they are not diffeo- morphisms). In a recent joint work with Welington de Melo (avail- able at arXiv:1303.3470) we proved that any two C3 critical circle maps with the same irrational rotation number of bounded type and with a unique critical point of the same odd criticality are conjugate to each other by a C1+? circle diffeomorphism, for some universal ? > 0. This geometric rigidity was conjectured in the early eight- ies (after several works of Feigenbaum, Kadanoff, Lanford, Rand, Shenker among others) and has promoted a lot of previous results in the real-analytic category (Swiatek, Herman, de Faria-de Melo, Yampolsky, Khanin-Teplinsky among others). In this talk we will discuss the main ideas of the proof. Geometric Rigidity of critical circle maps | 26 June 2013 |

Andrey Shilnikov (Georgia State University) | **Abstract:** We identify and describe the principal bifurcations of bursting rhythms in multi-functional
central pattern generators (CPG) composed of three neurons connected by fast inhibitory or excitatory synapses.
We develop a set of computational tools that reduce high-order dynamics in biologically relevant CPG models to low-dimensional return mappings that measure the phase lags between cells. We examine bifurcations of fixed points and invariant curves in such mappings as coupling properties of the synapses are varied. These bifurcations correspond to changes in the availability of the network's phase locked rhythmic activities such as periodic and aperiodic bursting patterns. As such, our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of, and switching between, motor patterns generated by various animals. Key bifurcations of bursting polyrhythms in three-cell central pattern generators
| 12 June 2013 |

Andrey Shilnikov (Georgia State University) | **Abstract:** TBA Chaos: stirred not shaken
| 11 June 2013 |

Mason Porter (Oxford University) | **Abstract:** I discuss "simple" dynamical systems on networks and examine how
network structure affects dynamics of processes running on top of
networks. I consider results based on "locally tree-like" and/or
mean-field and pair approximations and examine when they seem to work
well, what can cause them to fail, and when they seem to produce accurate
results even though their hypotheses are violated fantastically. I'll
also present a new model for multi-stage complex contagions--in which
fanatics produce greater influence than mere followers--and examine
dynamics on networks with hetergeneous correlations. (This talk discusses
joint work with James Gleeson, Sergey Melnik, Peter Mucha, JP Onnela, Felix Reed-Tsochas, and Jonathan Ward.) Cascades and Social Influence on Networks | 30 May 2013 |

Michel Crucifix (Université catholique de Louvain) | **Abstract:** Glacial-interglacial cycles have paced climate for over 3 million years. The phenomenon is thought to result from the interplay between non-linear internal dynamics and the extrenal forcing induced by the variations in the Earth's orbit and obliquity, commonly modelled as quasi-periodic functions.
Here we concentrate to a number of different simple determinstic models of ice ages, available in the form of small systems of ordinary differential equations. Our objective is to explore and explain the sensitivity of these models to initial conditions and parameters that has been previously discussed in the literature, in terms of dynamical system theory. To this end, we use a series of tools and concepts, including the section of the pullback attractor, the Lyapunov exponent, and the phase sensitivity exponents.
The most interesting behaviours appear to be linked to the emergence of strange nonchaotic attractors (SNA). The scenario corresponds to a negative long-term Lyaponov exponent, with sensitive dependence both to parameters and additive noise. This dependence is manifested in the form af trajectory shifts, which may be interpreted as a consequence of the non-smooth character of the attractor.
We discuss the implications of these results on our understanding of palaeoclimate dynamics and ability to predict future glacial cycles.
References:
T. Mitsui and K. Aihara, Dynamics between order and chaos in conceptual models of glacial cycles, published online in Climate Dynamics (available at http://link.springer.com/article/10.1007/s00382-013-1793-x
M. Crucifix, Why glacial cycles could be unpredictable ? accepted in Climate of the Past (available at http://arxiv.org/abs/1302.1492)
Glacial cycles and strange non-chaotic attractors
| 22 May 2013 |

Lev Lerman (Lobachevsky University of Nizhni Novgorod) | **Abstract:** I’ll discuss results obtained in our group in Nizhny Novgorod (former Gorky) mostly in seventies of XX century when studying smooth nonautonomous flows on smooth closed manifolds. The main classifying relation for this study was taken the uniform equivalence for two nonautonomous flows, more precisely, for their foliations into integral curves in the extended phase space. This allowed us to give a notion of structurally stable nonautonomous flows and find a natural class of flows for which the invariant of its uniform equivalency was found was proved they to be structurally stable. Also it gave a possibility to find connections between the structure of flows and topology of the ambient manifold (like Morse inequalities). One more application of these notions was the relation of uniform homotopy equivalence for the maps that classified diffeomorphisms and gave a source of constructing nonautonomous flows with various structure. Nonautonomous flows and uniform topology | 1 May 2013 |

Jason Gallas (Universidade Federal da Paraíba) | **Abstract:** The investigation of regular and irregular patterns in nonlinear
oscillators is an outstanding problem in physics and in all natural
sciences.
In general, regularity is understood as tantamount to periodicity.
However, there is now a flurry of works proving the
existence of ``antiperiodicity'', an unfamiliar type of regularity.
In this seminar, we present a report about the experimental
observation and numerical corroboration of antiperiodic oscillations.
In contrast to the isolated solutions presently known,
we show several examples of infinite hierarchies of antiperiodic waveforms that
can be tuned continuously and that form wide spiral-shaped
stability phases in the control parameter plane (i.e. in phase diagrams).
The waveform complexity increases towards the focal point common to all
spirals, a key hub interconnecting them all. Antiperiodic oscillations | 25 April 2013 |

Vered Rom-Kedar (Weizmann Institute) | **Abstract:** Simple models of the innate immune system teach us much about the
development of infections when the bone-marrow function is damaged by
chemotherapy. The results depend only on robust properties of the
underlying modeling assumptions and not on the detailed
models. Such models may lead to improved treatment
strategies for neutropenic patients [1,2,3,4].
[1] Roy Malka, Baruch Wolach, Ronit Gavrieli, Eliezer Shochat and Vered Rom-Kedar, Evidence for bistable bacteria-neutrophil interaction and its clinical implications J. Clin Invest. doi:10.1172/JCI59832, 2012. See also commentary .
[2] R. Malka and V. Rom-Kedar, Bacteria--Phagocytes Dynamics, Axiomatic Modelling and Mass-Action Kinetics, Mathematical Biosciences and Engineering, 8(2), 475-502, 2011.
[3] E. Shochat and V. Rom-Kedar; Novel strategies for G-CSF treatment of high-risk
severe neutropenia suggested by mathematical modeling, Clinical Cancer Research 14,
6354-6363, October 15, 2008.
[4] E. Shochat, V. Rom-Kedar and L. Segel; G-CSF control of neutrophils dynamics
in the blood, Bull. Math. Biology , 69(7), 2299-2338, 2007
The innate immune system: some theory, experiments and medical implications | 24 April 2013 |

Sergey Yakovenko (Weizmann Institute) | **Abstract:** The original Hilbert 16th problem about the limit cycles of polynomial planar vector fields stays open for over 110 years; its centennial history was full of dramatic turns, as exposed in the survey "Centennial History of the Hilbert 16th Problem" by Yulij Ilyashenko. About 50 years ago Ilyashenko himself suggested a problem about zeros of the Poincare-Pontryagin integral which describes bifurcation of limit cycles in perturbations of integrable (Hamiltonian) systems.
I will discuss various precise formulations of this problem and recent results by G. Binyamini, D. Novikov and the speaker giving explicit and existential upper bounds for the number of isolated zeros of Abelian and pseudoabelian integrals. The talk is intended for a broad audience. Semicentennial history of the Infinitesimal Hilbert 16th problem | 9 April 2013 |

Alejandro Passeggi (TU Dresden) | **Abstract:** We give an introduction to the rotation theory in the two dimensional torus which can be seen as the generalization of Poincaré theory in the circle, and present the following result:
"There exists an open and dense set of homeomorphisms homotopic to the identity (with respect the $C^0$ topology), such that the rotation set of its elements is a rational polygon". Rational Polygons as Rotation Sets of Generic Homeomorphisms
of the two Torus | 21 March 2013 |

Mike Field (Rice University
) | **Abstract:** For dynamicists, a network consists of interconnected dynamical systems (or "nodes").
Classical networks encountered in dynamics are synchronous: nodes all run on the
same clock and connectivity is fixed. However, most networks
encountered in contemporary science and technology are asynchronous. In particular,
biological networks, computer networks and distributed systems generally
are asynchronous: nodes may run on different clocks, connectivity may vary in
time and nodes may stop and then restart: the antithesis of 'analytic behaviour'
expected of solutions of smooth differential equations. In this talk, we describe
how a dynamicist might approach the definition and mathematics of asynchronous
networks as well as describe and illustrate some recent results and observations
about dynamics on asynchronous networks and possible mechanisms that allow them
to work efficiently. Asynchronous Networks
| 14 March 2013 |

Ivan Wong (University of Manchester) | **Abstract:** Piecewise smooth maps appear as models of various physical, economical and other systems. In such maps bifurcations can occur in when a fixed point (or periodic orbit) crosses or collides with the border between two regions of smooth behaviour. These bifurcations have little analogue in standard bifurcation theory for smooth maps and they are now known as border collision bifurcations. In this talk, we show that for piecewise smooth maps which allow area expansions, the dynamics of the system is not necessarily trivial. In particular, snap-back repellers and two-dimensional attractors can exist for appropriate parameter values. Border collision bifurcations – snap-back repellers and two-dimensional attractors | 7 March 2013 |

Thomas Bridges (University of Surrey ) | **Abstract:** The backbone of the talk is "modulation"; specifically what to modulate and how modulation generates geometry. The talk will be based on three examples. (1D modulation) how modulation gives a new viewpoint on elementary homoclinic bifurcation with curvature of modulation determining the coefficient of the nonlinear term. (2D modulation) the mythical origins of the KdV equation are given a new perspective, resulting in a universal form for emergence and how geometry of modulation determines the coefficients, and a dynamical systems argument determines the dispersion. Considering that the classical derivation of KdV is about the trivial solution, a by-product of the result is how to modulate the trivial solution! (3D modulation) The third example will show how modulation of periodic solutions leads to a sequence of multi-pulse planforms in PDEs like the Swift-Hohenberg equation. This theory is based on a new modulation equation in two space dimensions and time.
How modulation generates geometry in dynamical systems and nonlinear waves | 28 February 2013 |

Zeng Lian (Loughborough University) | **Abstract:** Lyapunov exponents play an important role in the study of the behavior of dynamical systems, which measure the
average rate of separation of orbits starting from nearby initial points. They are used to describe the local stability of orbits
and chaotic behavior of systems. Multiplicative Ergodic Theorem provides the theoretical fundation of Lyapunov exponents,
which gives the fundamental information of Lyapunov Exponents and their associates invariant subspaces.
In this talk, I will report the work on Multiplicative Ergodic Theorem (with Kening Lu), which is applicable to infinite dimensional
random dynamical systems in a separable Banach space. The system could be generated by, for example, random partial
differential equations.
Lyapunov exponents and Multiplicative Ergodic Theorem for random systems in a separable Banach space | 21 February 2013 |

Sofia Trejo (Warwick University) | **Abstract:** I will talk about the construction of complex box mappings with complex bounds for real analytic interval maps. More specifically, I will show that given a real analytic map and a point in its postcritical set it is possible to construct complex box mappings, associated to the real first return maps, with complex bounds for arbitrarily small scales.
In the case the map is a non-renormalizable polynomial (not necessarily real) with only hyperbolic periodic points, the complex-box mapping can be constructed using the Yoccoz puzzle. For real analytic maps we can not guarantee the existence of a Yoccoz puzzle. For this reason the construction of the box mapping and the prove of complex bounds in this case requires more work.
Complex bounds are fundamental for the prove of quasiconformal rigidity, renormalization results and ergodic properties. Complex Bounds for real analytic interval maps.
| 14 February 2013 |

Franco Vivaldi (Queen Mary) | **Abstract:** We consider a model of planar rotations subject to round-off, which leads to
dynamics on a lattice. We let the angle of rotation approach a low-order rational.
There is a non-smooth integrable Hamiltonian system, featuring a foliation by
polygonal invariant curves, which represents the limit of vanishing discretisation
of the space. We prove that, for sufficiently small discretization, a positive
fraction of those invariant curves survives, leading to a discrete space version
of the KAM scenario. The surviving curves are characterised in terms of congruences
and properties of Gaussian integers.
joint work with Heather Reeve-Black Near-Integrable behaviour in a system with discrete phase space | 7 February 2013 |

Vasso Anagnostopoulou
(Imperial College) | **Abstract:** We study a class of model systems which exhibit the full two step
scenario for the nonautonomous Hopf bifurcation, as proposed by Ludwig
Arnold. The specific structure of these models allows for a rigorous
and thorough analysis of the bifurcation pattern. In particular, we
show the existence of an invariant 'torus' splitting off a previously
stable central manifold after the second bifurcation point. A model for the nonautonomous Hopf bifurcation | 31 January 2013 |

John Lowenstein (New York University) | Renormalization of Piecewise Isometries | 24 January 2013 |

Ole Peters (London Mathematical Laboratory) | **Abstract:** Ensemble averages are frequently used as the basis for decision
theories in economics and games. However, they are often applied in
situations which do not correspond physically to ensembles. In many
common problems, such as that of an individual deciding how to invest
his wealth, it would be more appropriate to look at time averages.
The question of whether these two averages are identical leads to the
concept of ergodicity. In a non-ergodic system they may differ, making
it vital to know which is relevant. Economists have largely failed to
make this distinction and, as a result, are stuck with a confusing and
incomplete mathematical framework.
The problems caused by the inappropriate use of ensemble averages and
the failure to recognise non-ergodicity are illustrated by a famous
example: the St Petersburg Paradox. At best, these failings have
spurred the development of arbitrary and unjustifiable techniques,
such as utility theory. At worst, they have caused mathematical errors
to lie hidden in the economics literature for decades.
Proper acknowledgement of ergodicity and the role of time leads
naturally to a simple resolution of the St Petersburg Paradox. We wish
to explore how a deeper understanding of ergodic theory might shed
light on other foundational problems in economics. In particular, we
are keen to establish a standard nomenclature bringing together the
different definitions of ergodicity that exist in dynamical systems,
stochastic processes, and economics itself. Ergodicity in Economics
| 17 January 2013 |

Thomas Jordan (University of Bristol) | **Abstract:** This is joint work with Godofredo Iommi. In the setting of one dimension expanding maps the topological pressure function is well known to depend analytically on suitably defined permutations. The same is true when we move from expanding maps to suspension flows. In the case where the expanding map has countably many branches (for example the Gauss map) the pressure function has been studied by Sarig and Mauldin and Urbanski. In this case if the system is modeled by the full shift then when finite the pressure function varies analytically. We will look at the case when suspension flows over countable state maps are considered and give certain conditions under which the pressure varies analytically and show examples where there are phase transitions in the pressure function. Phase transitions for suspension flows | 10 January 2013 |

Phil Boyland (University of Florida) | **Abstract:** Given $\beta>0$, Renyi's beta-shift $Z_\beta$ encodes the collection of expansions base $\beta$ of all $x\in [0,1]$. The \textit{digit frequency vector} records the relative frequencies
of the various digits in the beta-expansion of a given x, and the set of all such vectors for a given beta is its \textit{digit frequency set}. We show that this set is always
compact, convex, $k-$dimensional (where $k \leq \beta \leq k+1$), and it varies continuously with $\beta$. When $k=2$ we give a complete description of the family of the digit frequency sets:
roughly, it looks like nested convex-set valued devil's staircases.
Considering the family of sets with the Hausdorff topology, the typical frequency set has countably infinite vertices with a single, completely irrational limit vertex. We then discuss how these results yield a near complete understanding of the rotation sets and their bifurcations of a family of two-torus homeomorphisms.
Rotation sets for beta-shifts and torus homeomorphisms | 13 December 2012 |

Peter Hazard (University of Warwick
) | **Abstract:** Roughly speaking, a modulus of stability is a number which is invariant under topological conjugacy. For unimodal maps there is a basic invariant - the kneading invariant - which determines the topological class of a unimodal map if the map is sufficiently smooth. However, it is known that in dimension two there are maps with infinitely many moduli. We will discuss what happens `in between', in the setting of Henon-like maps which are small perturbations of unimodal maps. This is joint work with M. Martens and C. Tresser. Infinite Moduli of Stability for Henon-like Maps | 6 December 2012 |

Anatoly Neishtadt (Loughborough University
) | **Abstract:** We study a classical billiard of charged particles in a strong non-uniform magnetic field. We provide an adiabatic description for skipping motion along the boundary of the billiard. We show that a sequence of many changes of regimes of motion from skipping to motion without collisions with the boundary and back to skipping leads to destruction of the adiabatic invariance and chaotic dynamics in a large domain in the phase space. This is a new mechanism of the origin of chaotic dynamics for systems with impacts. Destruction of adiabatic invariance for billiards in strong non-uniform magnetic field | 29 November 2012 |

Rajendra Bhansali (University of Liverpool) | Long memory properties of stochastic intermittency maps | 22 November 2012 |

Jens Marklof (University of Bristol ) | **Abstract:** Despite their simple geometry, billiards in polygons give rise to a rich variety of dynamical phenomena. One example is the asymptotic distribution of closed billiard trajectories, which is still only partially understood. In the present lecture I will discuss a different natural problem: the distribution of eigenfunctions of the Dirichlet Laplacian of a polygon--- is the L^2 mass of the eigenfunctions highly localized or equidistributed on the billiard domain? The lecture is based on joint work with Zeev Rudnick (Tel Aviv). Eigenfunctions of polygonal billiards
| 15 November 2012 |

Wolfram Just (Queen Mary, University of London) | **Abstract:** We apply time-delayed feedback control to stabilise unstable periodic
orbits of an amplitude-phase oscillator. The control acts on both, the
amplitude and the frequency of the oscillator.The model
has been introduced by Fiedler and Sch"oll as a counterexample for
the so called odd-number limitation of time-delayed feedback control.
A comprehensive bifurcation analysis in terms of the control phase and
the control strength reveals large stability regions of the target
periodic orbit, as well as an increasing number of unstable periodic
orbits caused by the time delay of the feedback loop. The theoretical
results are illustrated by an experimental realisation
of the time-delayed feedback scheme proposed by Sch"oll and Fiedler.
The experimental control performance is in quantitative agreement with
the bifurcation analysis. The results uncover some general features of
the control scheme which are deemed to be relevant for a large class of setups.
Pyragas-Schöll-Fiedler control | 8 November 2012 |

Gorka Zamora (Humboldt Universitaet) | **Abstract:** Despite the significant differences in the sizes of brains in the animal kingdom, there is increasing evidence that they all share similar modular and hierarchical organization. Modelling the brain activity to reproduce the observed functional networks brings important challenges to solve. Beyond the purely computational limitations, we also need to face theoretical issues before we are able to reproduce brain-like dynamics with certain degree of confidence. Complex brain networks: structure, modeling and function | 1 November 2012 |

Natalia Janson (Loughborough University) | **Abstract:** A single neuron is modelled as a highly non-linear dynamical system. Its crucial feature is excitability:
when there is no input, or the input is below a threshold, the neuron is silent, and when the input is above the threshold, the neuron
fires. The noise can play a highly counter-intuitive role in such systems: with the increase of
the amount of noise, the amount of order in the system grows. Thus, the noise-induced spiking in such neurons becomes almost regular at the optimal intensity of the stimulus. Many neurons are coupled together in a network by various means, to model a biological neural network.
Their collective behavior ranges from independent spiking, through partly and fully synchronised spiking, to the lack of any spiking, depending on the parameters of the coupling. Synchronized spiking in the brain is associated with epilepsy, Parkinsonian disease and tremor, so the ability to eliminate synchronization by
non-invasive weak stimulation could be essential in treating such conditions. It appears that in the model of the stochastic neural network it is possible
to partly control the collective behavior by a specially constructed feedback. Self-organisation and synchronization in stochastic neuron-like networks
| 24 October 2012 |

Jaap Eldering (Imperial College) | **Abstract:** Normally Hyperbolic Invariant Manifolds (NHIMs for short) are an
important tool to globally study perturbations of dynamical systems.
I will first recall what NHIMs are (simply put, these are
generalisations of hyperbolic fixed points) and then indicate how my
result on persistence of noncompact NHIMs generalizes the classical
compact case.
Noncompactness requires us to introduce the concept of Riemannian
manifolds of bounded geometry. These can be viewed as the class of
uniformly C^k manifolds. I will assume no detailed knowledge of
differential geometry, but illustrate this with images.
Standard analysis techniques and results can be adapted to this
setting. I will illustrate this with the example of constructing a
uniformly sized tubular neighborhood of a (noncompact) submanifold,
which is necessary to construct the persistent manifold. Normally Hyperbolic Invariant Manifolds the noncompact way | 18 October 2012 |

Yuzuru Sato (RIES, Hokkaido University) | **Abstract:** Interaction between deterministic chaos and stochastic randomness has been an important problem in nonlinear dynamical systems studies. Noise-induced phe- nomena are understood as drastic change of natural invariant densities by adding external noise to a deterministic dynamical systems, resulting qualitative transition of observed nonlinear phenomena. Stochastic resonance, noise-induced synchronization, and noise-induced chaos are typical examples in this scheme. The simplest mathematical model for problem is one-dimensional map stochastically perturbed by noise.
In this presentation, we discuss typical behavior of noised dynamical sys- tems based on numerically observed noise-induced phenomena in logistic map, Belousov-Zhabotinsky map and modified Lasota-Mackey map. Our observation indicates that (i) both noise-induced chaos and noise-induced order may coexist, and that (ii) asymptotical periodicity of densities varies according to noise amplitude. An application to time-series analysis of rotating fluid is also exhibited. Noise-induced phenomena in one-dimensional maps | 18 October 2012 |

Sebastian van Strien (Imperial College) | **Abstract:** One of the best known dynamical systems with intermittency behaviour is the well-known Pomeau-Manneville circle map $x\mapsto x+x^{1+\alpha} \mod 1$. This map has a neutral fixed point at $0$ which causes orbits to linger there for long periods. Nevertheless this map has always a physical measure: for $\alpha\ge 1$ it is the Dirac measure at $0$ while for $\alpha\in (0,1)$ it is absolutely continuous. It was also known for quite a while that this map is stochastically stable when $\alpha\ge 1$. In this talk I will discuss a result which implies that this map is also stochastic stable when $\alpha\in (0,1)$. (joint with Weixiao Shen)
Stochastic stability of expanding circle maps with neutral fixed points | 11 October 2012 |

Genadi Levin (Hebrew University) | **Abstract:** We consider an infinitely renormalizable map f_c with all its renormalizations of non-primitive (satellite) type. Accosiated to it is a combinatorial data: a sequence of (rational non-zero) rotation numbers {t_m=p_m/q_m} of the dividing fixed points of the renormalizations of f_c. Equivalently, the parameter c is a limit point of a sequence of "satellite bifurcations" with the "internal arguments" {t_m}. For example, the stationary sequence {t_m=1/2} corresponds (on the real line) to the famous Feigenbaum parameter. I explain a procedure (Douady, Hubbard, Sorensen), which shows that, if the sequence {t_m} tends to zero fast enough, then M is locally connected at c while J_c is not. Then I describe a framework allowing to get a class of explicit combinatorics, for which this effect occurs. For instance, any sequence {1/q_m} with q_{m+1}>2^{q_m} fits. Rigid non-locally connected Julia sets | 4 October 2012 |

Janosch Rieger (Goethe-Universitaet Frankfurt) | **Abstract:** The implicit Euler scheme for nonlinear ordinary differential inclusions was recently shown to possess favourable analytiv and convergence properties.
As in the ODE case, its performance is substantially better than that of the explicit Euler scheme if the underlying differential inclusion is stiff.
This effect is more pronounced than in the classical case, because the size of the explicit Euler images grows rapidly when stability is lost, which causes an exponential explosion of computational costs.
The spatial discretization of the implicit Euler scheme, however, is problematic, because its construction involves explicit knowledge of the modulus of continuity of the right-hand side. The semi-implicit Euler schemes presented in this talk overcome this problem. In addition, their performance is significantly better than that of the implicit Euler scheme. Semi-Implicit Methods for Differential Inclusions | 1 October 2012 |

Katsutoshi Shinohara (Pontifícia Universidade Católica do Rio de Janeiro) | **Abstract:** We consider (attracting) free semigroup actions on the interval with two generators.
It is known that, if those two generators are sufficiently C^2-close to the identity, then there is a restriction on the shape of the (forward) minimal set.
Namely, it must be the whole interval.
(This statement is not accurate.
I will give the precise statement in my talk.)
In this talk, I will explain that the similar argument fails in C^1-topology. On minimality of free-semigroup actions
on the interval C^1-close to the identity | 21 June 2012 |

Begoña Alarcón (University of Oviedo) | **Abstract:** We consider sufficient conditions to determine the global dynamics
for equivariant maps of the plane with a unique fixed point which is
also hyperbolic. When the map is equivariant under the action of a
subgroup of O(2), it is possible to describe the local dynamics and –
from this – also the global dynamics. In particular, if the group contains
a reflection, there is a line invariant by the map. We will see in
this seminar that this invariant line allows us to use results based on
the theory of free homeomorphisms of the plane to describe the global
dynamical behaviour. Otherwise, in the absence of reflections, equivariant
examples can be used to show that global dynamics may not follow
from local dynamics near the unique fixed point.
This is joint work with Isabel S. Labouriau (Centro de
Matematica, Universidade do Porto). Global dynamics for symmetric planar maps | 10 May 2012 |

Vasso Anagnostopoulou (Technical University of Dresden) | **Abstract:** We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. Replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both in a topological and a measure-theoretic setting. As an interesting new phenomenon, a dichotomy appears for the behaviour at the bifurcation point, which allows the bifurcation to be either 'smooth' (as in the classical case) or 'non-smooth'. Non-autonomous saddle-node bifurcations | 26 April 2012 |

Peter Hazard (University of Warwick) | **Abstract:** A well-know theorem of A. Katok states that if a sufficiently smooth diffeomorphism of a compact surface has positive topological entropy then it has a homoclinic point, and consequently infinitely many periodic orbits. A counterexample to this statement for homeomorphisms was later given by M. Rees who constructed a positive entropy homeomorphism of the 2-torus which was minimal.
I will discuss new elementary constructions of this type which replace minimality with the 'no periodic orbits' property, with some other related results. This is joint work with E. de Faria and C. Tresser. Periodic Points and Entropy on Surfaces | 15 March 2012 |

Sergey Gonchenko (University of Nizhny Novgorod) | **Abstract:** We study stable dynamics (both regular and chaotic) of the well-known mechanical system "celtic stone" (called sometimes as "celt" or "rattleback", or "wobblestone", or even "Russian stone"). Physically, it is a canoe-shaped rigid body with the curious property of spin asymmetry: it tends to have the stable vertical rotation in one direction only, independently on initial conditions for rotation. In the talk we try to explain this dynamical property by means of some mathematical idealization - the nonholonomic model of celtic stone. We study this model by numerical and qualitative methods and describe various stable regimes: permanent rotation, oscillations and, finally, chaotic dynamics. On regular and chaotic dynamics of "celtic stone" | 8 March 2012 |

Tiago Pereira (Imperial College) | **Abstract:** A remarkable property of Hermitian ensembles is their universal behavior, that is, once properly rescaled the eigenvalue statistics does not depend on particularities of the ensemble. According to the theory of random matrices, the eigenvalue correlations in Hermitian are given by the determinant of an integral kernel. The limit integral kernel is well known to be universal for standard models of Hermitian ensembles. The scenery for normal ensembles, despite of certain efforts in this direction remains undisclosed. We study the integral kernel of a certain ensembles of normal matrices, and as a corollary we obtain the universality of eigenvalue statistics. Universality in Normal Random Matrix Ensembles | 1 March 2012 |

Thomas Berger (Technical University Ilmenau) | **Abstract:** We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic
equations. The effects of perturbations in the leading coefficient matrix are investigated. A reasonable class of allowable perturbations is introduced. Robustness results in terms of the Bohl exponent and perturbation operator are presented. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples. On perturbations in the leading coefficient matrix of time-varying index-1 DAEs | 27 February 2012 |

Pablo Shmerkin (University of Surrey) | **Abstract:** The Hausdorff dimension of sets invariant under conformal dynamical systems can often be realized as the zero of certain natural pressure equation (going back to Bowen). This pressure is usually continuous as a function of the defining dynamics, in the appropriate topology, and hence so is the Hausdorff dimension of the invariant set.
The situation is dramatically more complicated in the non-conformal situation where, nevertheless, a subadditive version of the pressure equation, involving singular values of a matrix cocycle, is crucial. A natural question is therefore whether this subadditive pressure is also a continuous function of the dynamics (or, what is the same, of the associated cocycle). We resolve this in the affirmative in many important situations, in particular answering a question of Falconer and Sloan.
This is joint work with De-Jun Feng (Chinese University of Hong Kong). Continuity of subadditive pressure | 23 February 2012 |

Stefanella Boatto (Universidade Federal do Rio de Janeiro ) | The Poisson equation, the Robin function and singularities' dynamics:
an hydrodynamics approach
| 16 February 2012 |

Davoud Cheraghi (University of Warwick) | **Abstract:** The local, semi-local, and global dynamics of the complex quadratic polynomials $P_\alpha(z):=e^{2\pi i \alpha}z+z^2: \mathbb{C}\to \mathbb{C}$,
for irrational values of $\alpha$, have been extensively studied through various methods over the last decades. The difficulty comes from the interplay between the tangential movement created by the fixed point and the radial movement created by the critical point which brings in the arithmetic nature of $\alpha$. Using a renormalisation technique we analyse this interaction, and in particular, describe the topological behavior of the orbit of typical points under these maps. Typical orbits of complex quadratic polynomials with a neutral fixed point | 9 February 2012 |

André Caldas (Universidade de Brasilia) | Product type dynamical systems and the variational principle | 26 January 2012 |

Vassilis Rothos (University of Thessaloniki) | **Abstract:** The existence of quasi periodic travelling waves solutions in DNLS equation with nonlocal interactions and with polynomial type potentials will be considered. Calculus of variations and topological methods are employed to prove the existence of these type of solutions. Travelling waves in nonlocal lattice equations | 19 January 2012 |

Yizhar Or (Technion Israel) | **Abstract:** The motion of swimming microorganisms and robotic microswimmers is governed by low Reynolds number hydrodynamics where viscous effects dominate and inertial effects are negligible. The time-independence of Stokes equations augmented by structural geometric symmetries can lead to a reversing symmetry which governs the swimming dynamic equations. This is demonstrated in the talk for three different dynamic models: fixed-shape swimmer near a wall, Purcell’s three link swimmer near a wall, and torque control of the three-link swimmer. It is shown that breaking the geometric symmetries can lead to asymptotic stability of solutions which have clear physical meaning. Experimental results on macro-scale robotic swimmers will be reported, and the relation of the results to observations from swimming microorganisms’ behavior will be discussed.
Bio: Dr. Yizhar Or earned his PhD in 2007 at the Technion, Israel, in the field of robot dynamics. During 2007-2009 he was a postdoctoral scholar with Prof. Richard Murray in the Dept. of Control and Dynamical Systems at Caltech, USA, funded by Fulbright Program and the Israeli Science Foundation (Bikura Program). He is currently a Senior Lecturer in the Faculty of Mechanical Engineering at the Technion, Israel. His research interests are in nonlinear dynamics, mechanics and control of locomotion, with applications to robotics and biology. Reversing symmetry and dynamic stability in low-Reynolds-number swimming | 24 November 2011 |

Jürgen Knobloch (TU Ilmenau) | **Abstract:** Homoclinic snaking refers to the sinusoidal ``snaking'' continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium $E$ and a periodic orbit $P$. Along this curve the homoclinic orbit performs more and more windings about the periodic orbit. Typically this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of $E$ and $P$. We show how the snaking behaviour depends on the signs of the Floquet multipliers of $P$. Nonreversible Homoclinic Snaking | 24 November 2011 |

Vered Rom-Kedar (Weizmann Institute) | **Abstract:** A geometrical model which captures the main ingredients governing
atom-diatom collinear chemical reactions is proposed. This model is
neither near-integrable nor hyperbolic, yet it is amenable to analysis
using a combination of the recently developed tools for studying systems
with steep potentials and the study of the phase space structure near
a center-saddle equilibrium. The nontrivial dependence of the reaction
rates on parameters, initial conditions and energy is thus qualitatively
explained. Conditions under which the phase space transition state
theory assumptions are satisfied and conditions under which these fail
are derived. Extensions of these ideas to other impact-like systems and to
other models of reactions will be discussed.
Joint works w L. Lerman and M. Kloc.
A saddle in a corner - a model of atom-diatom chemical reactions | 18 November 2011 |

Dmitry Turaev (Imperial College) | **Abstract:** We prove that the attractor of the 1D quintic complex Ginzburg-
Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-
Landau type equations. We provide an analytic proof for the existence of two-
soliton configurations with Shilnikov-type chaotic temporal behavior, and construct solutions
which are close to a grid of such chaotic soliton pairs, with every pair in
the grid well spatially separated from the neighboring ones for all time. The
temporal evolution of the well-separated multi-soliton structures is described
by a weakly coupled lattice dynamical system (LDS) for the coordinates and
phases of the solitons. We develop a version of normal hyperbolicity theory
for the weakly coupled LDS’s with continuous time and establish for them
the existence of space-time chaotic patterns reminiscent of the Sinai-Bunimovich
chaos in discrete-time LDS’s. While the LDS part of the theory may be of
independent interest, the main difficulty concerns with
lifting the space-time chaotic solutions of the LDS back to the initial PDE.
The equations we consider are space-time autonomous, i.e. we impose
no spatial or temporal modulation which could prevent the individual solitons
in the grid from drifting towards each other and destroying the well-separated
grid structure in a finite time. We however manage to show that the set of
space-time chaotic solutions for which the random soliton drift is arrested is
large enough, so the corresponding space-time entropy is strictly positive.
This is a joint work with S. Zelik. Space-time chaos in Ginzburg-Landau equations | 10 November 2011 |

Sergey Zelik (University of Surrey) | **Abstract:** We discuss various aspects of the theory of exponential attractors for autonomous and non-autonomous dissipative systems generated by PDEs. The possible extensions of this theory to the case of random dynamical system and the new difficulties arising here will be also discussed. Finally, some recent results on the convergence of stochastic exponential attractors to the limit deterministic one when the amplitude of white noise tends to zero will be presented. Exponential attractors: from non-autonomous to random dynamics | 3 November 2011 |

Julian Newman (Imperial College) | **Abstract:** In traditional free body analysis, unknown contact forces are calculated by using Newton's Laws to form equations of motion, which can then be solved for these unknown contact forces. However, if there are more contact forces than there are equations of motion, then this will generally be impossible. This is particularly problematic when friction is involved, because it can give rise to the situation that one cannot, by traditional means, determine whether a system undergoes sliding or sticking. However, recent numerical simulations carried out by engineers in Germany suggest that the problem might be resolvable in the case of a disc in contact with two frictionable walls. The potential resolution comes from "regularising" the walls - that is, taking into account that the walls are not completely rigid. In this talk, results of an analytical study of the engineers' results are presented. Tracking Variations of Indeterminable Contact Forces between a Disc and a Frictionable Wedge: Part II | 3 November 2011 |

Julian Newman (Imperial College) | **Abstract:** In traditional free body analysis, unknown contact forces are calculated by using Newton's Laws to form equations of motion, which can then be solved for these unknown contact forces. However, if there are more contact forces than there are equations of motion, then this will generally be impossible. This is particularly problematic when friction is involved, because it can give rise to the situation that one cannot, by traditional means, determine whether a system undergoes sliding or sticking. However, recent numerical simulations carried out by engineers in Germany suggest that the problem might be resolvable in the case of a disc in contact with two frictionable walls. The potential resolution comes from "regularising" the walls - that is, taking into account that the walls are not completely rigid. In this talk, results of an analytical study of the engineers' results are presented. Tracking Variations of Indeterminable Contact Forces between a Disc and a Frictionable Wedge: Part I | 27 October 2011 |

Philipp Düren (University of Augsburg) | **Abstract:** When considering control systems in discrete time one can define invariant control sets as sets of "maximal controllability". On the other hand, the discussion of (also time-discrete) random diffeomorphisms (as for example used by H. Zmarrou, A. J. Homburg et al.) often uses the notion of stationary measures. We will work with a random diffeomorphism, called System A. When interpreting the stochastic noise of A as a arbitrary control we obtain a control system B. We will see that the supports of stationary measures of A correspond bijectively to the invariant control sets of B. This is extendable to open systems as well. Invariant control sets and stationary measures | 20 October 2011 |

Gabor Kiss (University of Exeter) | **Abstract:** In many applications the rate of change of state variables depends on their states at prior times. When these processes are assumed to be deterministic, they are modelled by functional differential equations.
In the simplest cases only one, time invariant time lag is considered.
However, equations with multiple delays offer richer dynamics, thus they are of mathematical interest and potential models of real-world problems with complex oscillations. We present results on the coexistence of periodic solutions to equations with multiple delays.
Furthermore, we report on the existence of pullback attractors to equations with multiple time-varying delays. A model for exchange-rate fluctuations is considered as a motivating example. Ideas for possible future work are outlined. Oscillating solutions of functional differential equations | 20 September 2011 |

Shangjiang Guo (Hunan University) | **Abstract:** This talk deals with the existence, monotonicity, uniqueness, asymptotic behavior, and nonlinear stability of travelling wavefronts for temporally delayed spatially discrete reaction-diffusion equations. Based on the combination of the weighted energy method ad the Green function technique, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near the negative infinity regardless of the magnitude of time delay. Wavefronts in Discrete Reaction-Diffusion Equations with Nonlocal Delayed Effects | 14 September 2011 |

Sergey Gonchenko (University of Nizhny Novgorod) | On Newhouse regions with mixed dynamics | 28 March 2011 |

Lev Lerman (University of Nizhny Novgorod) | Integrable Hamiltonian systems: structure and bifurcations | 21 March 2011 |

Thorsten Huels (University of Bielefeld) | Computing dichotomy projectors and Sacker-Sell spectra in discrete
time dynamical systems | 2 March 2011 |

Boumediene Hamzi (Imperial College) | Model Reduction of Nonlinear Control Systems in Reproducing Kernel
Hilbert Space | 1 February 2011 |

Martin Rasmussen (Imperial College) | Morse decompositions of nonautonomous and set-valued dynamical systems
| 25 January 2011 |

Chris Warner (Imperial College) | Long-time Asymptotics for a Classical Particle Interacting with a Scalar Wave Field
| 16 December 2010 |

Konstantinos Kourliouros (Imperial College) | Typical Singularities of Constrained Systems II
| 25 November 2010 |

Konstantinos Kourliouros (Imperial College) | Typical Singularities of Constrained Systems I | 18 November 2010 |

Lina Avramidou (University of Surrey) | **Abstract:** Given a real-valued function f defined on the phase space of a dynamical system, ergodic optimization is the study of the orbits that maximize the ergodic f-time-average. It turns out that this is equivalent in maximizing the space average over all invariant probability measures. In this talk, we examine the question of maximization of non-conventional ergodic averages along square iterates for the doubling map and fairly simple functions f. Ergodic Optimization along the squares | 23 June 2010 |

Leonid Bunimovich (Georgia Tech) | **Abstract:** A natural question on how a position of a 'hole" in a phase space
seems to be never studied. The answer is interesting by itself and
demonstrates that the dynamical systems theory can make finite time
predictions of dynamics.
Some new results for dynamical networks and even for Markov chains are
obtained along these lines as well. Open Systems and Dynamical Networks | 14 June 2010 |

Andrei Vladimirov (WIAS Berlin) | Dissipative localised structures of light and their interaction. | 27 April 2010 |

Oliver Jenkinson (QMUL) | How to maximize long-term happiness? - Ergodic optimization of utility functions. | 24 March 2010 |

Fritz Colonius (Augsburg) | **Abstract:** Control of digitally connected dynamical systems is a subject which has recently found considerable interest. In this talk, an abstract approach is presented intending to specify minimal data rates for control tasks.
It is based on a concept which is motivated by the notion of topological entropy in the theory of dynamical systems.
Entropy-like notions in control | 16 March 2010 |

Paulo Ruffino (UNICAMP) | **Abstract:** Given two complementary distributions in the tangent bundle
of a manifold,
we find conditions to factorize an stochastic flow into a diffusion in
the (infinite dimensional)
Lie group of diffeomorphisms which preserve one distribution
(horizontal), composed with a process
in the Lie group of diffemorphisms which preserve the other
distribution (vertical). This decomposition generalizes
previous approach, e.g. using coordinate maps, by Ming Liao and others. Decomposition of stochastic flows along complementary distributions | 20 January 2010 |

Joseph Rosenblatt (University of Illinois at Urbana-Champaign) | **Abstract:** We want to characterize which sequences of whole numbers $(n_m)$ admit a weakly mixing transformation $\tau$ such that $\tau$ is rigid along $(n_m)$, and which times $(n_m)$ admit a weakly mixing transformation $\tau$ such that $\tau$ is not recurrent along $(n_m)$. These questions are in opposition, but also related. The necessary properties are a mix of sparsity and combinatorial, and/or algebraic, structure of the sequence. Examples and counterexamples, as well as some general results, will be described. This talk is based on joint current work with V. Bergelson (Ohio State), A. del Junco (Toronto), and M. Lemanczyk (Torun). Rigidity and Recurrence for Dynamical Systems | 13 January 2010 |

Vered Rom-Kedar (The Weizmann Institute) | Stability in high dimensional steep repelling potentials and the Boltzmann ergodic hypothesis.
| 18 November 2009 |

Wael Bahsoun (Loughborough University) | Random maps, skew-products and existence of invariant measures | 27 October 2009 |

Christian Poetzsche (TU Munich) | Discrete Dynamics and Nonlinear Analysis: A Commensal Relationship!
| 7 October 2009 |

Guillaume Defrance (Jussieu) | Using Matching Pursuit for estimating mixing time within Room Impulse Responses | 8 July 2009 |

Ole Peters (Imperial College) | On time and risk | 8 June 2009 |

John Roberts (UNSW) | **Abstract:** In a program joint with F Vivaldi (Queen Mary), we have shown that
structural properties of discrete dynamical systems leave a universal
signature on the reduced dynamics over finite fields (analogous to the
division of quantum spectral statistics into those of certain random
matrix ensembles). I will briefly review previous results, then consider
reversible rational maps, i.e. those maps in d-dimensional space that
can be written as the composition of 2 rational involutions. We study
the reduction of such rational maps to finite fields and look to study
the proportion of the finite phase space occupied by cycles and by
aperiodic orbits and the length distributions of such orbits. We find
that the dynamics of these low-complexity highly deterministic maps has
some universal (i.e. map-independent) aspects. The distribution is well
explained using a combinatoric model that averages over an ensemble of
pairs of random involutions in the finite phase space. Universal Period Distribution for Reversible Rational Maps over
Finite Fields | 21 May 2009 |

Alexandre Rodrigues (University of Porto) | Switching near a Heteroclinic Network of Rotating Nodes | 7 May 2009 |

Renato Vitolo (Exeter) | The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: resonance `bubbles' and routes to chaos | 24 March 2009 |

Jaap Eldering (Utrecht) | The Perron method for invariant fibrations | 24 March 2009 |

Peter Giesl (University of Sussex) | **Abstract:** The basin of attraction of equilibria or periodic orbits of an ODE can be determined through sublevel set of a Lyapunov function. To construct such a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the ODE, a linear PDE is solved approximately using Radial Basis Functions. Error estimates ensure that the approximation is a Lyapunov function.
For the construction of a Lyapunov function it is necessary to know the position of the equilibrium or periodic orbit. A different method to analyse the basin of attraction of a periodic orbit without knowledge of its position is Borg's criterion. The sufficiency and necessity of this criterion in different settings will be discussed. Determination of the Basin of Attraction of Equilibria and Periodic Orbits | 24 March 2009 |

Andrey Shilnikov (Atlanta) | TBC | 19 March 2009 |

Sergey Gonchenko (University of Nizhny Novgorod) | Attractors and repellers in reversible maps with heteroclinic tangencies | 17 March 2009 |

Tony Samuel (St Andrews) | Dynamics and noncommutative geometry | 11 March 2009 |

Pawel Pilarczyk (Universidade do Minho, Portugal) | A method for automatic classification of global dynamics in multi-parameter systems | 10 March 2009 |

Tomás Lázaro (Universitat Politécnica de Catalunya) | **Abstract:** Tchebycheff systems share most of the nice properties satisfied by the
set of polynomials of a given degree and, in this sense, they become
its natural extension.
Despite the fact that there exist algorithms to build such a sets,
there are no many examples in the literature. In this work we prove
that a suitable type of functions forms a Tchebycheff system. This
family of functions uses to appear when one deals with Abelian
Integrals in some concrete problems associated to the Weak 16th
Hilbert's Problem.
This is a joint work (in progress) with A. Gasull and J. Torregrosa,
Departament de Matemátiques, Universitat Autónoma de Barcelona,
Spain. Using Tchebycheff systems to
estimate the number of zeroes of some Abelian Integrals | 3 March 2009 |

Thomas Jordan (Bristol) | Where is a topological conjugacy differentiable? | 24 February 2009 |

Peter Kloeden (Frankfurt) | Random attractors and the preservation of synchronization in the presence of noise (AMMP colloquium)
| 17 February 2009 |

Henk Bruin (Surrey) | Li-Yorke chaos and Cantor attractor of interval maps | 17 February 2009 |

Corinna Ulcigrai (Bristol) | **Abstract:** We consider a class of area-preserving (locally Hamiltonian) flows on a surface of genus g. We are interested in their ergodic properties, especially mixing: it turns out that the presence/absence of mixing depends on the type of fixed points. We proved that the presence of centers in a generic such flow is enough to create mixing. Recently we showed that if the flow has only saddles, it is generically not mixing, but weakly mixing. The results uses the flows representation as suspensions over interval exchange transformations and the study of deviations of Birkhoff averages over interval exchanges. Mixing properties of area-preserving flows on surfaces
| 10 February 2009 |

Martin Andersson (ENS, Paris) | Bifurcations of physical measures | 3 February 2009 |

Christian Rodrigues (University of Aberdeen) | Emergent attractors for weakly dissipative systems | 27 January 2009 |

Jeroen Lamb (Imperial College) | **Abstract:** In many examples of classical mechanics, reversibility (playing a film of the dynamics yielding a physically realistic event) arises simultaneously with the occurrence of a Hamiltonian structure of the equations of motion. In fact reversibility has often be identified as a useful tool to prove results for mechanical systems, and indeed many important results can be proven based on the assumption of a Hamiltonian structure or the presence of a time-reversal symmetry. In this talk I will review our understanding of the differences and similarities between reversible and Hamiltonian dynamics. Reversible versus Hamiltonian dynamical systems | 21 January 2009 |

Isabel Rios (Universidade Federal Fluminense) | Twisted Horseshoes | 20 January 2009 |

Rafael Ortega (Granada) | A property of stable fixed points of area-preserving maps | 15 January 2009 |

Martin Rasmussen (Imperial College) | Three different approaches to the approximation of nonautonomous invariant manifolds (AMMP Colloquium) | 13 January 2009 |

Oliver Butterley (Imperial College) | Transfer operators for suspension semiflows | 28 November 2008 |

Ian Melbourne (University of Surrey) | Validity of the 0-1 test for chaos | 26 November 2008 |

Mark Holland (University of Exeter) | Extreme Events in Chaotic Dynamical Systems | 19 November 2008 |

Sergey Zelik (University of Surrey) | Dissipative dynamics in large and unbounded domains: attractors, entropies and space-time chaos | 12 November 2008 |

Oleg Makarenkov (Imperial College London) | Nonsmooth bifurcation theory and mechanics (AMMP Colloquium) | 11 November 2008 |

Anatoly Neishtadt (Loughborough University) | Separatrix crossings in slow-fast Hamiltonian systems | 5 November 2008 |

Vassili Gelfreich (University of Warwick) | Fermi acceleration in non-autonomous billiards
| 29 October 2008 |

Lev Lerman (University of Nizhny Novgorod) | **Abstract:** Let a 2 d.o.f. Hamiltonian system have an equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearisation matrix. Such a degeneration is generically met in one parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. We prove the Lyapunov stability of the equilibrium when some coefficient of the 4th order normal form is positive (the equilibrium is unstable, if this coefficient is negative). The result is known since 1977, but both proofs having appeared are incorrect. The proof is based on the KAM theory and uses a work with Weierstrass elliptic functions, estimates of power series and methods of the theory of dynamical systems. On stability at the Hamiltonian Hopf
Bifurcation | 15 October 2008 |

Bob Rink (Free University Amsterdam) | Continuum equations for the Fermi-Pasta-Ulam chain | 8 October 2008 |

Juergen Knobloch (TU Ilmenau) | Snaking near heteroclinic cycles | 17 September 2008 |

Zalman Balanov (Netanya) | Application of Twisted Equivariant Degree to Symmetric Hopf
Bifuraction | 7 July 2008 |

Wieslaw Krawcewicz ( Alberta) | Variational Problems and Gradient Equivariant Degree
| 7 July 2008 |

Ramon Driesse (Amsterdam) | Essential Asymptotic Stability Of A Homoclinic Cycle | 10 June 2008 |

Jens Rademacher (CWI) | Unfolding heteroclinic networks of equilibria and periodic orbits using Liapunov-Schmidt reduction | 26 February 2008 |

Alexander Plakhov (University of Wales - Aberystwith) | **Abstract:** A body moves through a rarefied medium of point particles. The particles hit the body in the absolutely elastic manner and do not mutually interact. Find the body, out of a given class of admissible bodies, such that the force of resistance of the medium to its motion is minimal. This is the Newton problem of minimal resistance. We will discuss several generalizations of this problem, as well as related questions of billiards theory and Monge-Kantorovich optimal mass transportation. Some possible applications (retroreflectors, Magnus effect) will also be discussed. Billiards, Newtonian aerodynamics, and optimal mass transportation | 22 January 2008 |

Marc Georgi (Berlin) | Bifurcations of Travelling Waves in Lattice Differential Equations | 22 January 2008 |

Michael Field (University of Houston) | Global dynamics and recurrence in (small) coupled dynamical systems. | 20 November 2007 |

James Meiss (University of Colorado) | Dynamics and Bifurcations in Volume Preserving Maps | 13 November 2007 |

Ian Melbourne (University of Surrey) | Statistical and probabilistic properties of dynamical systems
| 30 October 2007 |

Andrew Stuart (University of Warwick) | Probabilistic Inverse Problems in Differential Equations | 16 October 2007 |

Michael Jakobson (University of Maryland) | **Abstract:** For quadratic-like families f we are reviewing constructions in the phase space and in the parameter space, and estimating the measure of parameter values such that f have absolutely continuous invariant measures. Estimating measures in the parameter space | 4 July 2007 |

Kie van Ivanky Saputra (La Trobe University Melbourne) | Alternative walk in the parameter space | 20 June 2007 |

Raul Ures (Montevideo, Uruguay) | **Abstract:** Abstract. In a previous work we proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stable ergodic diffeomorphism are dense among the partially hyperbolic ones. In this talk we shall address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we shall give the first examples of manifolds in which al l conservative partially hyperbolic diffeomorphisms are ergodic. The
talk will be based in a joint work with Federico Rodriguez Hertz and Mar ́ıa A. Rodriguez Hertz. Partial hyperbolicity and ergodicity in dimension three | 12 June 2007 |

Phil Boyland (University of Florida) | **Abstract:** A covering space is a way of "unwinding" a manifold, transforming loops into translations. Lifting a dynamical system to a cover allows one to understand how the dynamicswraps around the manifold by examining how lifted orbits translate in the cover. After introducing the rotation set, the talk will focus on a kind of "hyper-transitivity" where a given map has a dense orbit when lifted to the largest Abelian cover. Roughly speaking, this implies that a single orbit explores repeatedly all the ways of traversing loops in the manifold. I will begin with the relatively simple circle
case and then focus mainly on iterated surface homeomorphisms. Unraveling Dynamics via covering spaces | 16 May 2007 |

Henk Bruin (Surrey) | Equilibrium States for One-Dimensional Maps | 19 February 2007 |

Jorge Freitas (Porto) | Statistical stability for some non-hyperbolic systems | 19 February 2007 |

Oscar Bandtlow (Queen Mary University of London) | Invariant measures for analytic maps with unbounded distortion | 19 February 2007 |

Jose' Alves (Porto) | Mixing rates and hyperbolic structures for partially hyperbolic diffeomorphisms | 19 February 2007 |

Todd Young (Ohio & Warwick) | Intermittency in Unimodal Maps | 19 February 2007 |

Y. Sinai (Princeton University) | Limiting Behaviour for Large Frobenius Numbers | 31 January 2007 |

Oliver Butterley (Imperial College) | Transfer operators for Anosov flows | 19 January 2007 |

Thomas Jordan (Warwick) | Randomly perturbed self-affine sets | 19 January 2007 |

Ian Morris (Manchester) | Estimation of the maximum ergodic average | 19 January 2007 |

Neil Dobbs (Paris Orsay) | Markov maps with integrable return times | 19 January 2007 |

Mike Todd (Surrey) | Hofbauer towers in ergodic theory | 19 January 2007 |

Paulo Ruffino (Campinas, Brazil) | **Abstract:** Consider a cocycle of random orientation preserving diffeomorphisms on the
circle
$f(\omega)$, based on a probability space $(\Omega, F,
P)$, with an ergodic shift $\theta: \Omega --> \Omega$. We present an
ergodic theorem for the rotation number $R_{(f,\theta)}$ of the
composition of the random sequence $(f(\theta^n \omega))$. If
$R_{(f,\theta)}$ is irrational, we look for conditions for the existence
of a random (measurable)
homeomorphism $h(\omega)$ which provides the cocycle conjugacy with rotation:
f(\omega)= h^{-1}(\theta \omega)\circ R_{(f(\omega))} \circ h(\omega).
Yet, we investigate the existence of this cocycle conjugacy for a
stochastic flow (non structurally stable) in $S^1$. Conditions for a random version of Denjoy Theorem | 5 December 2006 |

Guido Gentile (Rome) | **Abstract:** Existence of lower-dimensional tori in quasi-integrable
Hamiltonian systems can be proved by various means.
In this talk I shall discuss a method based on
renormalization group techniques, with the aim of
showing that the Bryuno condition arises as a very
natural Diophantine condition to be imposed on the
frequency vectors of the persisting invariant tori. Renormalization group for lower-dimensional tori under the Bryuno condition | 28 November 2006 |

Guillaume James (Toulouse) | **Abstract:** The propagation of nonlinear waves in spatially discrete media gives rise to
interesting phenomena. An effect of spatial discreteness can be the formation
of "hot spots" where vibrational energy remains localized without dispersion.
The study of "discrete breathers" (time-periodic and spatially localized
oscillations) in nonlinear Hamiltonian oscillator chains provides an
interesting mathematical framework for studying this phenomenon.
In this talk we consider the Fermi-Pasta-Ulam lattice, which consists in a chain
of particles nonlinearly coupled to their nearest neighbours (here the chain is
of infinite extent). When the system admits a hard interaction potential, the
existence of weakly localized breathers has been predicted by Tsurui in the
70s, and the existence of strongly localized ones has been suggested by Sievers
and Takeno in the 80s, both studies using formal approximations.
A delicate mathematical question is to determine if such approximate solutions
correspond to exact breather solutions for the oscillator chain. As we shall
see, this question leads us to analyze the dynamics of an infinite-dimensional
nonlinear map, whose linear part is an unbounded operator. Thanks to good
spectral properties, the local dynamics of the map is shown to be
finite-dimensional, which allows us to conclude on the existence or
nonexistence of breathers in the small amplitude limit. The question of the existence of "breathers" in the Fermi-Pasta-Ulam model | 28 November 2006 |

Richard Sharp (Manchester) | **Abstract:** We discuss ways of comparing lengths of pairs of closed geodesics on negatively curved surfaces. We focus on pairs where the difference of the lengths lies in an interval which is allowed to shrink. This is inspired by problems of pair correlations for eigenvalues in quantum chaos. If time permits, we will discuss analogous results for ergodic sums over hyperbolic systems. (Joint work with Mark Pollicott.)
Refs:
M Pollicott and R Sharp, Correlations for pairs of closed geodesics, Invent. math. 163 (2006), 1-24.
M Pollicott and R Sharp, Distribution of ergodic sums for hyperbolic maps, in "Representation Theory, Dynamical Systems, and Asymptotic Combinatorics" (ed. V Kaimanovich and A Lodkin), American Mathematical Society (2006). Pair correlations and length spectra on negatively curved sufaces | 31 October 2006 |

Juergen Knobloch (TU Ilmenau) | Chaotic behavior near homoclinic points with quadratic tangency | 3 October 2006 |

Martijn van Noort | **Abstract:** We study a Hamiltonian parametrically forced pendulum system with two parameters, and show that for a large range of parameter values the system has a Cantor family of invariant tori in an annulus around its lower equilibrium, corresponding to quasiperiodic oscillations of the pendulum. The parametric forcing need not be very small, as long as it is smaller than the force of gravity, and the pendulum is short enough. Invariant oscillatory tori in the forced pendulum | 22 June 2006 |

Yakov Pesin (Penn State University) | Is chaotic behavior typical among dynamical systems? | 20 June 2006 |

Anatole Katok (Penn State University) | **Abstract:** We present an overview and some new applications of the
approximation by conjugation method introduced by Anosov and the second
author more than thirty years ago \cite{AK}. Michel Herman made
important contributions to the development and applications of this method
beginning from the construction of minimal and uniquely ergodic diffeomorphisms
jointly with Fathi in \cite{FH} and continuing with exotic invariant
sets of rational maps of the Riemann sphere \cite{H3}, and the construction of
invariant tori with nonstandard and unexpected behavior in the
context of KAM theory \cite{H1, H2}.
Recently the method has been experiencing a revival. Some of the new
results presented in the paper illustrate variety of uses for tools available
for a long time, others exploit new methods, in particular possibility of
mixing in the context of Liouvillean dynamics discovered by the first author
\cite{F1, F2}. Liouvillean phenomena in dynamics and ergodic theory | 14 June 2006 |

Tobias
Jaeger (University of Erlangen) | General pattern in the formation of strange non-chaotic attractors (AMMP Colloquium)
| 25 April 2006 |

Frank Schilder (University of Bristol) | **Abstract:** A famous phenomenon in circle-maps and synchronisation problems
leads to a two-parameter bifurcation diagram commonly referred
to as the Arnol'd tongue scenario. One considers a perturbation
of a rigid rotation of a circle, or a system of coupled oscillators.
In both cases we have two natural parameters, the coupling strength
and a detuning parameter that controls the rotation number/frequency
ratio. The typical parameter plane of such systems has Arnol'd
tongues with their tips on the decoupling line, opening up into
the region where coupling is enabled, and in between these Arnol'd
tongues, quasi-periodic arcs. In this talk we present unified
algorithms for computing both Arnol'd tongues and quasi-periodic
arcs for both maps and ODEs. The algorithms generalise and improve
on the standard methods for computing these objects. We illustrate
our methods by numerically investigating the Arnold tongue scenario
for two examples from electrical engineering: a parametrically
forced network and a system of coupled Van der Pol oscillators. Computing Arnol'd tongue scenarios | 12 April 2006 |

Cristina Ciocci (Imperial College London) | Steady states for the tippe top | 5 April 2006 |

Stavros Komineas (University of Cambridge) | **Abstract:** Quasi-one-dimensional solitons that occur in an elongated
Bose-Einstein condensate (BEC) are described by
a nonlinear Schroedinger equation (NLS).
These become unstable at high particle density.
Within a nonlinear Gross-Pitaevskii model
we study a basic mode of instability and the corresponding
bifurcation to genuinely three-dimensional axisymmetric vortex rings.
We calculate their profiles and examine their
dependence on the velocity of propagation along a cylindrical trap.
At sufficiently high velocity, the vortex ring
transforms into an axisymmetric soliton. We also calculate
the energy-momentum dispersions and show that a Lieb-type mode appears
in the excitation spectrum for all particle densities.
We further study interactions of solitons and vortex rings
in a cylindrical BEC by simulating their head-on collisions.
The results are compared against collisions of solitons in the NLS,
and also against the dynamics of solitary waves
in the three-dimensional homogeneous Bose gas.
We discuss a related recent experiment
[Ginsberg et al. Phys. Rev. Lett. 94, 040403 (2005)]. Solitons and vortex rings in a cylindrical Bose-Einstein condensate
| 22 March 2006 |

Martijn van Noort (Imperial College London) | **Abstract:** In this talk we employ KAM theory to rigorously investigate the transition between quasiperiodic and chaotic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices, modelled by a parametrically forced Duffing equation that describes the spatial dynamics of the condensate. We will show the existence of KAM tori for lattices of arbitrary size, that is, for shallow-well, intermediate-well, as well as deep-well potentials. Hence one obtains a large measure of quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, with rotation number proportional to the amplitude. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers. Bose-Einstein Condensates in Optical Lattices and Superlattices | 22 March 2006 |

Gerton Lunter (University of Oxford) | **Abstract:** To study the behaviour of Hamiltonian dynamical systems, singularity
theory is often useful in reducing the system to a simple normal form.
However, although singularity theory guarantees the existence of an
appropriate normalizing coordinate transformation, in applications it
is often desirable to be able to construct this transformation, for
instance to pull back bifurcation curves to original parameters. In this
talk I will show how Groebner basis techniques can be modified to give
efficient algorithms for computing normalizing transformations. Groebner bases and constructing normalizing transformations | 8 March 2006 |

Ana Paula Dias (University of Porto) | **Abstract:** In the analysis of stability in bifurcation problems it is often
assumed that the (appropriate reduced) equations are in normal
form. In the presence of symmetry, the truncated normal form is
an equivariant polynomial map. Therefore, the determination of
invariants and equivariants of the group of symmetries of the
problem is an important step. In general, these are hard problems
of invariant theory, and in most cases, they are tractable only
through symbolic computer programs. Nevertheless, it is desirable
to obtain some of the information about invariants and equivariants
without actually computing them. In this work we obtain formulas
for the number of linearly independent homogeneous invariants or
equivariants for Hopf bifurcation degree by degree in terms of
characters and we show that they are effectively computable in
several concrete examples. This information allows to draw some
predictions about the structure of the bifurcations. Invariants, Equivariants and Characters in Symmetric Bifurcation Theory | 3 March 2006 |

Alastair Rucklidge (University of Leeds) | Quasipatterns in surface waves (Applied Colloquium) | 9 November 2004 |

Bjorn Sandstede (University of Surrey) | Dynamics of coherent structures in oscillatory media (Applied Colloquium) | 26 October 2004 |

Alan Champneys (University of Bristol) | Rock, rattle and slide; towards a bifurcation theory for piecewise smooth systems | 26 October 2004 |

Michael Field (Imperial/University of Houston) | Geometry, symmetry and bifurcation (Applied Colloquium) | 5 October 2004 |

Stefanella Boatto (McMaster University) | Periodic solutions of Euler equations on the sphere | 5 October 2004 |

Fima Dinaburg | On statistical mechanics model built on sand | 23 June 2004 |

Victor Planas-Bielsa (INLN Nice) | Leibniz manifolds and Lyapunov stability of Poisson equilibria | 4 June 2004 |

John Elgin (Imperial College) | On the computation of multifractal spectra from time series data | 2 June 2004 |

Cristina Stoica (University of Surrey) | Relative Equilibria of Systems with Configuration Space Isotropy | 2 June 2004 |

Marco Antonio Teixeira (University of Campinas) | Invariant varieties for discontinuous vector fields | 10 May 2004 |

Konstantinos Efstathiou (University of Dunkerque) | Metamorphoses of Hamiltonian systems with symmetries | 27 April 2004 |

Mike Field (University of Houston) | **Abstract:** Recently Stewart, Golubitsky and coworkers have formulated a
general theory of networks of coupled cells.
Their approach depends on groupoids, graphs, balanced equivalence
relations and 'quotient networks'. We present a combinatorial
approach to coupled cell systems. While largely equivalent to that of Stewart et al.,
our approach is motivated by ideas coming from analog computers, is directed towards
eventual applications in engineering, and avoids abstract algebraic formalism. Combinatorial Dynamics | 24 March 2004 |

Odo Diekmann (University of Utrecht) | A crash course in physiologically structured population models (applied colloquium) | 17 February 2004 |

Rob Beardmore (Imperial College) | Some toy models for the evolution of disease | 17 February 2004 |

James Montaldi (UMIST) | **Abstract:** In the first part of the talk I will describe some recent results (joint
work with Mark Roberts and Frederic Laurent-Polz) on the stability of
configurations of point vortices on the sphere. In the second I will
describe in the general context of symmetric Hamiltonian systems the
different stability transitions that occur. Point vortices and stability transitions in symmetric Hamiltonian systems | 4 February 2004 |

Gerald Moore (Imperial College) | Floquet theory as a computational tool | 4 February 2004 |

Rob Beardmore (Imperial College) | An index-2 Kronecker normal form and singularities of DAEs | 21 January 2004 |

Vassilis Rothos (Queen Mary) | Bifurcations of travelling breathers in the discrete NLS equation. | 21 January 2004 |

Dmitrii Sadovskii (University of Dunkerque) | Qualitative analysis of internal molecular dynamics | 19 January 2004 |

Franz Gaehler (University of Stuttgart) | Spaces of tilings and their topology | 3 December 2003 |

Oliver Jenkinson (Queen Mary) | Sturmian orbits in ergodic optimization
| 3 December 2003 |

Uwe Grimm (Open University) | Shelling of planar tilings with N-fold symmetry | 3 December 2003 |

Jean-Paul Allouche (University of Paris-Sud, Orsay) | Iteration of continuous real maps, non-integer bases, and a fractal set of sequences | 3 December 2003 |

Edmund Harriss (Imperial College) | Canonical substitution tilings | 3 December 2003 |

Jens Marklof (Bristol) | Ergodic theory and the distribution of n^2 alpha mod 1 | 19 November 2003 |

Roland Zweimueller (Imperial) | Invariant measures for generalized induced transformations | 19 November 2003 |

Henk Bruin (Surrey) | Existence of invariant densities for unimodal maps without growth conditions. | 19 November 2003 |

Peter Ashwin (University of Exeter) | What is a random attractor and how do I know when I've got one? (Applied Colloquium) | 4 November 2003 |

Marcelo Viana (IMPA, Rio de Janeiro) | Homoclinic Bifurcations and Fractal Invariants for High Dimensional Dynamical Systems (Applied Colloquium) | 14 October 2003 |

Vassili Gelfreich (University of Warwick) | Homoclinic orbits near strong resonances in Hamiltonian systems | 14 October 2003 |

Ale Jan Homburg (University of Amsterdam) | Multiple homoclinic orbits in conservative and reversible systems | 14 October 2003 |

Henk Broer (University of Groningen) | **Abstract:** (joint work with Richard Cushman and Francesco Fass\`{o}):
The classical Kolmogorov Arnold Moser (KAM) theory deals with
persistence of Lagrangean invariant tori in nearly integrable
Hamiltonian systems. The persistent tori have Diophantine frequencies
and hence are parametrized over a nowhere dense set of almost full
measure. The KAM theory provides conjugacies between the unperturbed,
integrable tori and the perturbed, nearly integrable tori which are
smooth in the sense of Whitney. However, these conjugacies in the action
space are only locally defined.
We obtain a global version of the Hamiltonian KAM theorem for invariant
Lagrangean tori by glueing together the local KAM conjugacies with help
of a partition of unity. In this way we find a global Whitney smooth
conjugacy between a nearly-integrable system and an integrable one. The
global conjugacy at once is an isomorphism of torus bundles.
This leads to preservation of geometry, which allows us to define all
the nontrivial geometric invariants like monodromy or Chern classes of
an integrable system also for nearly integrable systems.
This result is relevant for semiclassical versions of nearly integrable
systems. Geometry of KAM tori for nearly integrable Hamiltonian systems | 3 October 2003 |

Kevin Webster (Imperial College) | Heteroclinic cycle bifurcation in 3D reversible vector fields | 3 October 2003 |

Carlangelo Liverani (University of Rome) | **Abstract:** I consider a one-parameter family of area-preserving smooth maps that cross a non-uniformly hyperbolic situation into an elliptic one. I prove that exponentially close to such a family there are maps with positive metric entropy. Birth of an elliptic island in a chaotic sea | 16 September 2003 |

Yongluo Cao (Suzhou University, P.R. China) | The basin problems of attractors | 5 September 2003 |

Bob Rink (University of Utrecht) | **Abstract:** At Los Alamos in 1954, nobel prize winner Fermi, computer expert Pasta and
mathematician Ulam performed a numerical experiment to study the ergodic
properties of a one-dimensional continuum. They discretised this continuum
by considering a lattice of material elements, each of which interacts with
its neighbours. Statistical mechanics postulates that nonlinearities in the
interparticle forces will then make the equations of motion ergodic such
that the lattice reaches a thermal equilibrium. The numerical integration
was therefore amazing, as it actually turned out that the lattice behaved
quasi-periodically. This paradox is known as the Fermi-Pasta-Ulam problem.
One possible and generally excepted explanation for this observation is
based on the Kolmogorov-Arnol'd-Moser (KAM) theorem, which assures
quasi-periodicity under certain restrictive conditions. But proofs that
these conditions are satisfied have been absent for 50 years. In this talk
I will present the first complete proof in this direction, which makes use
of Birkhoff normal forms, symmetries and number theory. Dynamical and
geometric properties of the equations of motion are emphasized.
Geometry and dynamics in the Fermi-Pasta-Ulam lattice | 5 September 2003 |

Andrew Torok (University of Houston) | **Abstract:** We show that Axiom A flows are generically stably mixing on each
of their nontrivial basic sets. This is a consequence of our results concerning generic stable transitivity of smooth suspensions over a hyperbolic base (the fiber being either a compact Lie group or R^n).
Stable mixing for hyperbolic flows, and stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets | 19 June 2003 |

Willy Govaerts | MatCont: an interactive Matlab package for dynamical systems, continuation and bifurcation | 22 May 2003 |

Jaroslav Stark (Imperial College) | Invariant Sets for Quasiperiodically Forced Maps. | 2 April 2003 |

Gerhard Keller (Erlangen) | Some remarks on strange nonchaotic attractors for quasiperiodically driven systems | 2 April 2003 |

Jacques Fejoz (Paris VI) | The problem of the stability of the solar system | 17 February 2003 |

Gabriel Paternain (Cambridge) | An introduction to boundary rigidity for Lagrangian submanifolds and Aubry-Mather theory | 27 November 2002 |

Richard Sharp (Manchester) | Periodic orbits of Anosov flows and homology. | 27 November 2002 |

Ian Stewart, FRS (Warwick) | Dynamics on Networks | 20 November 2002 |

Henrik Jensen (Imperial) | An introduction tothe Theory of Self-Organizing Critialities | 13 November 2002 |

Kevin Webster (Imperial) | The Kupka-Smale Theorem for Differential Equations | 16 October 2002 |

Vincent Lynch (Warwick) | Decay of correlations | 9 October 2002 |

Juergen Knobloch (TU Ilmenau) | Bifurcation from degenerate homoclinic orbits | 12 July 2002 |

Oleg Stenkin (Imperial College) | Conservative and non-conservative behaviour in Newhouse regions | 12 July 2002 |

Mike Field (University of Houston) | Stable transitivity and ergodicity for compact abelian extensions over general hyperbolic basic sets | 29 May 2002 |

Mark Holland (University of Manchester) | Slowly mixing systems and intermittency maps | 29 May 2002 |

Kenneth Meyer (University of Cincinnati) | **Abstract:** We consider the evolution of the stable and unstable manifolds of an
equilibrium point of a Hamiltonian system of two degrees of freedom which
depends on a parameter, $\nu $. The eigenvalues of the linearized system are
complex for $\nu < 0$ and pure imaginary for $\nu > 0$. Thus, for $ \nu < 0 $
the equilibrium has a two-dimensional stable manifold and a two-dimensional
unstable manifold, but for $ \nu > 0 $ these stable and unstable manifolds are
gone.
If the sign of a certain term in the normal form is positive then for small
negative $\nu$ the stable and unstable manifolds of the system are either
identical or must have transverse intersection. Thus, either
the system is totally degenerate or the system admits a suspended Smale
horseshoe as an invariant set. Evolution of invariant manifolds | 15 May 2002 |

Joao da Rocha Medrado (Autonomous University of Barcelona) | Symmetric singularities of reversible vector fields of R^{n} | 15 May 2002 |

Isabel Rios (Universidade Federal Fluminense) | **Abstract:** We study
one-parameter families $(f_\mu)_{\mu\in [-1,1]}$
of two dimensional diffeomorphisms
unfolding critical saddle-node horseshoes (say at $\mu=0$) such that
$f_\mu$ is hyperbolic for negative $\mu$.
We
describe the dynamics at some
isolated secondary bifurcations which appear in the sequel of
the unfolding of the initial
saddle-node bifurcation.
We construct two classes of open sets of such arcs.
For the first class,
we
exhibit a collection of parameter
intervals $I_n$, $I_n\subset (0,1]$,
converging to the saddle-node parameter, $I_n\to 0$,
such that the topological entropy of $f_\mu$ is a constant $h_n$
in $I_n$
and $h_n$ is an increasing sequence.
So, for parameters in $I_n$, the topological entropy
is upper bounded by the entropy of the initial saddle-node
diffeomorphisms. This illustrates the following intuitive
principle: a critical cycle of an attracting saddle-node horseshoe
is a destroying dynamics bifurcation. In the second class,
the entropy of $f_\mu$ does not depend monotonely on the parameter $\mu$.
Finally, when the saddle-node horseshoe is not an attractor, we
prove that the entropy may increase after the bifurcation. Critical saddle-node horseshoes: bifurcations and entropy | 30 April 2002 |

Oleg Stenkin (Imperial College) | About boundaries of intervals of hyperbolicity
at homoclinic Omega-explosion | 30 April 2002 |

Jeroen Lamb (Imperial College) | Normal form theory for linear reversible equivariant vector fields | 21 March 2002 |

Keith Briggs (BTexact Technologies) | Simultaneous Diophantine approximation and linearization of
C^{2} maps | 6 February 2002 |

Mauricio Barahona (Imperial College) | Synchronization in small-world systems | 6 February 2002 |

Oleg Kozlovski (University of Warwick) | Real Koebe Lemma | 30 January 2002 |

Shaun Bullett (Queen Mary) | Sturmian sequences and holomorphic correspondences | 30 January 2002 |

Thomas Wagenknecht (TU-Ilmenau) | Homoclinic orbits to degenerate equilibria in reversible systems | 16 January 2002 |

Claudio Buzzi (UNESP-Rio Preto) | Hamiltonian vector fields with symplectic time-reversing symmetry. | 16 January 2002 |

Miguel Mendes (University of Surrey) | Some recent developments and open questions on the dynamics of piecewise
isometries (slides) | 16 January 2002 |

Ben Mestel (Exeter University) | A golden mean functional recurrence | 12 December 2001 |

Adam Epstein (Warwick University) | **Abstract:** The moduli space of all quadratic rational maps up to M\"obius conjugacy
is isomorphic to ${\Bbb C}^2$. It is possible, and also useful, to regard
one of the coordinate axes as the moduli space of quadratic polynomials;
the Mandelbrot set, parametrizing the quadratic polynomials with connected
Julia set, thereby lies in this slice. Nearly twenty years ago, Douady
conjectured that the rational maps in the central portion of the moduli
space of quadratic rational maps might be understood as {\em matings} of
pairs of quadratic polynomials. The proposed construction is purely
topological: one glues filled-in Julia sets back-to-back along
complex-conjugate prime ends to obtain a branched cover of the sphere.
Work of Tan Lei and Mary Rees shows that under favorable circumstances,
the resulting branched cover is topologically conjugate to an essentially
unique quadratic rational map. According to Milnor, mating is an
interesting operation because it possesses none of the usual good
properties: it is not injective, surjective, continuous, or even
everywhere defined. We will survey recent results concerning these issues
- in particular, the discontinuity of mating. Matings of Quadratic Polynomials | 12 December 2001 |

Marcelo Viana (IMPA and College de France) | Deterministic products of matrices,
in Dynamics and other places | 7 December 2001 |

Konstantin Mishaikov (Georgia Institute of Technology) | Rigorous Computations for Infinite Dimensional
Dynamics. | 28 November 2001 |

Raymond Hide (Imperial College) | Nonlinear quenching of current fluctuations in a self-exciting dynamo | 28 November 2001 |

Oliver Jenkinson (Queen Mary and Westfield College, London) | Cohomology classes of dynamically non-negative c^k functions | 7 November 2001 |

Matt Nicol (Surrey University) | Statistical properties of compact group extensions
of chaotic systems | 7 November 2001 |

David Broomhead (UMIST) | Dynamical models of digital channels - from the sublime to the ridiculous | 24 October 2001 |

Franco Vivaldi (Queen Mary) | Hamiltonian round-off errors | 24 October 2001 |

Ian Melbourne (University of Surrey) | Ginzburg-Landau theory of transitions in spatially extended systems | 10 October 2001 |

Edgar Knobloch (University of Leeds) | New type of complex dynamics in the 1:2 spatial resonance | 10 October 2001 |

Fernando Sanchez-Salas | Markov towers for hyperbolic systems | 23 May 2001 |

Omri Sarig (Warwick) | **Abstract:** We describe a new method for estimating the correlation functions for
equilibrium measures for countable Markov shifts in situations when the
tail of the first return time function to some partition set is O(1/n^b)
for b>2. Under some aperiodicity condition, this method allows one to
determine the second order asymptotics of the iterates
of the transfer operator and therby obtain improved estimates for the
``rate of convergence to equilibrium''. These estimates are strong enough
to yield sharp upper AS WELL AS LOWER bounds for the correlation
functions, and show that at least in the polynomial case, LS Young's upper
estimates are sharp. Polynomial Lower Bounds for Rates of Decay of Correlations | 23 May 2001 |

Adam Epstein | **Abstract:** The moduli space of all quadratic rational maps up to M\"obius conjugacy is isomorphic to ${\Bbb C}^2$. It is possible, and also useful, to regard one of the coordinate axes as the moduli space of quadratic polynomials; the Mandelbrot set, parametrizing the quadratic polynomials with connected Julia set, thereby lies in this slice. Nearly twenty years ago, Douady conjectured that the quadratic rational maps in the central portion of moduli space might be understood as {\em matings} of pairs of quadratic polynomials. The proposed construction is purely topological: one glues filled-in Julia sets back-to-back along complex-conjugate prime ends to obtain a branched cover of the sphere. Work of Tan Lei and Mary Rees shows that under favorable circumstances, the resulting branched cover is topologically conjugate to an essentially unique quadratic rational map. According to Milnor, mating is an interesting operation because it possesses none of the usual good properties: it is not injective, surjective, continuous, or even everywhere defined. We will survey recent results concerning these issues. Matings of quadratic polynomials | 24 April 2001 |

Henk Bruin (Groningen) | Maximal automorphic factors and interval dynamics | 18 April 2001 |

Jose Alves (University of Oporto) | Stochastic Dynamics | 18 April 2001 |

Y. Puri | Arithmetic of numbers of periodic points | 20 March 2001 |