Oskar Weinberger: Realisations of heteroclinic networks in coupled cell systems

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 objects 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 my thesis and this presentation. I give a brief introduction to heteroclinic dynamics and coupled cell systems, and in particular to concepts relevant for the existence and realisation of heteroclinic networks. The focus will then be on the realisation of a two node heteroclinic cycle in a three cell network. The existence of this realisation has previously been proved, but as for most instances of heteroclinic networks in coupled cell systems there has been lacking explicit vector field constructions. Such an explicit construction is presented in the form of a polynomial vector field that robustly realises the heteroclinic cycle. It is also a largely unexplored phenomenon that a realisation of a heteroclinic network will typically induce additional equilibria or other invariant sets. For the heteroclinic cycle and cell network above, I show that a large class of realisations always have such so called forced equilibria.