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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