Cluster and group synchronization in delay-coupled networks

TL;DR

This paper analyzes the stability of synchronized states in delay-coupled networks with multiple groups or clusters, using a master stability approach and symmetry properties to simplify stability evaluation.

Contribution

It introduces a symmetry-based method to assess stability of group and cluster synchronization in complex delay-coupled networks with diverse local dynamics.

Findings

Master stability function exhibits discrete rotational symmetry.

Eigenvalue spectra of coupling matrices show similar symmetry.

Method applied to semiconductor lasers and neuronal models.

Abstract

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.