HETEROCLINIC NETWORKS IN COUPLED CELL

We give an intrinsic deenition of a heteroclinic network as a ow invariant set that is indecomposable but not recurrent. Our deenition 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 both structure of a heteroclinic network as well as dynamics on and near the network. In particular, we introduce a notion of`depth' for a hetero-clinic 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 nd a variety of complicated, yet robust, dynamics in simple polynomial vector elds that possess symmetries. For this model system, we nd and prove the existence of depth two networks involving connections between heteroclinic cycles and equilibria, and study bifurcations of such structures.

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