## Abstract

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,
e-mail: mikefield@gmail.com or Michael.J.Field@rice.edu

Professor Mike Field

Department of Mathematics

Rice University

6100 S Main St

Houston

TX 77005-1892