Cluster singularity: the unfolding of clustering behavior in globally coupled Stuart-Landau oscillators

TL;DR

This paper explores how cluster patterns form and transition in ensembles of Stuart-Landau oscillators, revealing a special cluster singularity point where bifurcations collapse and transitions occur.

Contribution

It introduces the concept of a cluster singularity in coupled Stuart-Landau oscillators, detailing the transition mechanisms between cluster states and synchronization.

Findings

Cluster states emerge in minimal networks.

Increasing oscillators causes crowding of 2-cluster states.

A cluster singularity leads to a continuous transition from clusters to synchronization.

Abstract

The ubiquitous occurrence of cluster patterns in nature still lacks a comprehensive understanding. It is known that the dynamics of many such natural systems is captured by ensembles of Stuart-Landau oscillators. Here, we investigate clustering dynamics in a mean-coupled ensemble of such limit-cycle oscillators. In particular we show how clustering occurs in minimal networks, and elaborate how the observed 2-cluster states crowd when increasing the number of oscillators. Using persistence, we discuss how this crowding leads to a continuous transition from balanced cluster states to synchronized solutions via the intermediate unbalanced 2-cluster states. These cascade-like transitions emerge from what we call a cluster singularity. At this codimension-2 point, the bifurcations of all 2-cluster states collapse and the stable balanced cluster state bifurcates into the synchronized solution supercritically. We confirm our results using numerical simulations, and discuss how our conclusions apply to spatially extended systems.