Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators

TL;DR

This paper establishes quantitative stability conditions for cluster synchronization in networks of heterogeneous Kuramoto oscillators, highlighting the roles of network weights, frequency differences, and intra-cluster coupling.

Contribution

It provides the first explicit conditions on network parameters and oscillator properties that guarantee stable cluster synchronization in heterogeneous Kuramoto networks.

Findings

Cluster synchronization stability depends on intra- vs inter-cluster coupling strength.

Distinct natural frequencies promote stable cluster formation.

Homogeneous intra-cluster dynamics enhance synchronization robustness.

Abstract

In this paper we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected network. Cluster synchronization is at the basis of several biological and technological processes; yet the underlying mechanisms to enable cluster synchronization of Kuramoto oscillators have remained elusive. In this paper we derive quantitative conditions on the network weights, cluster configuration, and oscillators' natural frequency that ensure asymptotic stability of the cluster synchronization manifold; that is, the ability to recover the desired cluster synchronization configuration following a perturbation of the oscillators' states. Qualitatively, our results show that cluster synchronization is stable when the intra-cluster coupling is sufficiently stronger than the inter-cluster coupling, the natural frequencies of the oscillators in distinct clusters are sufficiently different, or, in the case of two clusters, when the intra-cluster dynamics is homogeneous. We illustrate and validate the effectiveness of our theoretical results via numerical studies.