Synchronization for Networks of Globally Coupled Maps in the Thermodynamic Limit

Abstract: We study a network of finitely many interacting clusters where each cluster is a collection of globally coupled circle maps in the thermodynamic (or mean field) limit. The state of each cluster is described by a probability measure, and its evolution is given by a self-consistent transfer operator. A cluster is synchronized if its state is a Dirac measure. We provide sufficient conditions for all clusters to synchronize and we describe setups where the conditions are met thanks to the uncoupled dynamics and/or the (diffusive) nature of the coupling. We also give sufficient conditions for partially synchronized states to arise -- i.e. states where only a subset of the clusters is synchronized -- due to the forcing of a group of cluster on the rest of the network. Lastly, we use this framework to show emergence and stability of chimera states for these systems.