In Synchronization: Theory and Application (eds. A. Pikovsky and Y. Maistrenko) Kluwer: Dordrecht (2003) 5-23.

Cycling attractors of coupled cell systems and dynamics with symmetry

Peter Ashwin(1) A.M.Rucklidge(2) Rob Sturman(2).
(1) School of Mathematical Sciences, Laver Building,
University of Exeter, Exeter EX4 4QE, UK

(2) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK

Abstract. Dynamical systems with symmetries show a number of atypical behaviours for generic dynamical systems. As coupled cell systems often possess symmetries,these behaviours are important for understanding dynamical effects in such systems. In particular the presence of symmetries gives invariant subspaces that interact with attractors to give new types of instability and intermittent attractor. In this paper we review and extend some recent work (Ashwin, Rucklidge and Sturman 2002) on robust non-ergodic attractors consists of cycles between invariant subspaces, called `cycling chaos' by Dellnitz et al. (1995).

By considering a simple model of coupled oscillators that show such cycles, we investigate the difference in behaviour between what we call free-running and phase-resetting (discontinuous) models. The difference is shown most clearly when observing the types of attractors created when an attracting cycle loses stability at a resonance. We describe both scenarios - giving intermittent stuck-on chaos for the free-running model, and an infinite family of periodic orbits for the phase-resetting case. These require careful numerical simulation to resolve quantities that routinely get as small as 10^-1000.

We characterise the difference between these models by considering the rates at which the cycles approach the invariant subspaces. Finally, we demonstrate similar behaviour in a continuous version of the phase-resetting model that is less amenable to analysis and raise some open questions.

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