Multi-cluster dynamics in coupled phase oscillator networks

TL;DR

This paper investigates how specific coupling functions in identical, globally coupled phase oscillators can produce stable multi-cluster states and heteroclinic networks involving three or more clusters, expanding understanding of complex synchronization patterns.

Contribution

It demonstrates the existence of coupling functions that generate robust heteroclinic networks with three or more clusters, clarifying the link between coupling functions and multi-cluster dynamics.

Findings

Coupling functions can produce stable multi-cluster heteroclinic networks.

Existence of examples with three or more nontrivial clusters.

Transverse stability can be independently varied from tangential stability.

Abstract

In this paper we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function $g(\varphi)$ (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N=6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability.