In the presence of symmetries or invariant subspaces, dynamical structures for attractors in dynamical systems can become very complicated owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible.

This paper introduces and discusses an instructive example of an ODE where one can observe and analyse robust cycling behaviour. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rossler system), and/or saddle equilibria.

For this model, we distinguish between cycling that include phase resetting connections (where there are only a finite number of connecting trajectories) and more general non-phase resetting cases where there is an infinite number, or even a continuum, of connections. We discuss the instability of this cycling to resonances of Lyapunov exponents and conjecture, based on numerical results, that phase resetting cycles typically lead to stable periodic orbits at instability whereas more general cases may give rise to 'stuck on' cycling.

Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence simulation of phase-resetting and connection selection.

For preprint, email: mikefield@gmail.com

Professor Mike Field
Department of Mathematics
Imperial College
London SW7 2AZ