About Patterns of Synchrony in Coupled Cell Networks

A coupled cell system is a dynamical system distributed over the nodes (cells) of a network. Each cell is an individual dynamical system (which we are going to assume continuous) and the coupling structure of the network indicates the interactions between those cell dynamics. One of the key aspects in the theory of coupled cell networks concerns the existence of synchrony subspaces – subspaces defined in terms of equalities between cell coordinates which are flow-invariant. Synchrony subspaces (flow-invariant subspaces) can have a major impact on both global and local dynamics and are important from the point of view of the study of that dynamics. Surprisingly, synchrony subspaces are independent of the specific individual dynamics at the nodes and are determined only by the network structure. That is, all the coupled cell systems admissible by a given network structure share the same structure of patterns of synchrony. In my talk, I will review recent concepts and results concerning and related with synchrony subspaces in coupled cell networks. The talk includes joint work with Peter Ashwin (University of Exeter), Ana Dias (Univer- sity of Porto), Flora Ferreira (University of Porto), Mike Field (Rice University, Imperial College), Marty Golubitsky (The Ohio State University), Maria Leite (University of South Florida) and Haibo Ruan (Universität Hamburg).