## Abstract

We give an intrinsic definition of a heteroclinic network as
a flow invariant set that is indecomposable but not recurrent.
Our definition covers many previously
discussed examples of heteroclinic behavior. In addition,
it provides a natural framework for discussing
cycles between invariant sets more complicated
than equilibria or limit cycles. We allow for cycles that connect chaotic
sets (cycling chaos) or heteroclinic cycles
(cycling cycles). Both phenomena can
occur robustly in systems with symmetry.
We analyze the structure of a heteroclinic network
as well as dynamics on and near the network.
In particular, we introduce a notion of `depth' for a
heteroclinic network (simple cycles between equilibria have depth
one), characterize the connections and discuss issues of attraction,
robustness and asymptotic behavior near a network.

We consider in detail a system of nine coupled cells where
one can find a variety of complicated, yet robust, dynamics in
simple polynomial vector fields that possess symmetries.
For this model system, we find and prove the existence of
depth two networks involving connections between heteroclinic
cycles and equilibria, and study bifurcations
of such structures.

For preprint,
email: mikefield@gmail.com

Professor Mike Field

Department of Mathematics

Imperial College

London SW7 2AZ