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