Synchronization for Networks of Globally Coupled Maps in the Thermodynamic Limit

TL;DR

This paper investigates synchronization phenomena in networks of coupled circle maps in the thermodynamic limit, providing conditions for full and partial synchronization, and exploring the emergence of chimera states.

Contribution

It introduces a framework using probability measures and transfer operators to analyze synchronization and chimera states in large networks of coupled maps.

Findings

Sufficient conditions for complete synchronization of all clusters.

Conditions for partial synchronization states to occur.

Demonstration of stable chimera states in the network.

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.