Near-integrability in Hamiltonian dynamical systems

In recent years there has been much progress in understanding complex phenomena in ordinary and partial differential equations through the study of dynamical systems. In particular, the development of theories that take into account essential structural properties (such as symmetry, coupled cell network, and Hamiltonian structure) of dynamical systems has been very effective in providing a theoretical basis for understanding complicated phenomena such as pattern formation and symmetry breaking in fluid flows and mechanics. While the study of discrete (reversing) symmetries in generic dynamical systems and the theory of integrable Hamiltonian systems are well established, the importance of discrete (reversing) symmetries in Hamiltonian systems is less well-established, in particular in relation to near-integrability. The main aims are twofold:

To develop (symmetry and/or coupled cell network) criteria that lead to near-integrability of reversible equivariant Hamiltonian systems, in particular integrability of finite order Birkhoff normal forms.
To study generic dynamical properties of near-integrable reversible equivariant Hamiltonian dynamical systems, exploring the interaction between the geometry of the integrable approximation on the one hand and dynamical systems theory on the other hand.

The theory will involve the study of changes in dynamics when external parameters are varied (bifurcation theory), including the occurrence of complicated dynamical behaviour (chaos). In particular, we will focus on periodic solutions, invariant tori (KAM theory), and homoclinic and heteroclinic cycles (and associated hyperbolic dynamics). A number of typical applications including the Fermi-Pasta-Ulam chain, many-particle systems, fluid mechanics, molecular dynamics, and classical mechanics are the main driving forces behind our research and we aim to develop and apply our theory alongside these applications. Although mainly focussing on classical systems, we also intend to explore the consequences of our findings in quantum mechanics, in particular in the context of molecular dynamics.

Work in progress with Bob Rink (Imperial)