Cycling Attractors of Coupled Cell Systems and Dynamics with Symmetry

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 (1995).