Realisations of heteroclinic networks in coupled cell systems

Abstract: In the theory of dynamical systems, heteroclinic networks are invariant objects in phase space with network structure, consisting of invariant sets (nodes) and connecting trajectories between them (edges). These are typically not robust dynamical phenomena, but can appear robustly for dynamical systems with network structure - so called coupled cell systems -due to the presence of certain synchrony related invariant subspaces. This link between networks in phase space and networks of dynamical systems is the topic of this thesis. Examples and results from the literature on the existence and construction of heteroclinic networks in coupled cell systems are presented and reviewed, focusing on heteroclinic realisation: how can coupled cell systems be constructed that support a given heteroclinic network? We seek to find explicit vector fields for such realisations, of which there are relatively few examples in the literature, and provide a polynomial vector field for a particular heteroclinic network and coupled system. Finally, we state and prove a theorem on the existence of additional equilibrium points for realisations of this heteroclinic network in such systems.