Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks

TL;DR

This paper introduces a comprehensive method to identify and analyze all possible cluster synchronization patterns in Laplacian-coupled networks, combining computational group theory with stability analysis, validated through experiments.

Contribution

It develops a novel approach to find and evaluate the stability of all cluster synchronization patterns in Laplacian networks using computational group theory.

Findings

Validated predictions with electro-optic experiments

Identified all possible cluster synchronization patterns

Provided a stability evaluation method for each pattern

Abstract

Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted in order to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based upon the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. The understanding of how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe here a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically-equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an electro-optic experiment on a 5 node network that confirms the synchronization patterns predicted by the theory.