Two-cluster solutions in an ensemble of generic limit-cycle oscillators with periodic self-forcing via the mean-field

TL;DR

This paper investigates two-cluster solutions in ensembles of Stuart-Landau oscillators with nonlinear global coupling, revealing complex bifurcation structures and various synchronization states near a Hopf bifurcation.

Contribution

It introduces a reduction to effective equations that enable linear stability analysis of cluster solutions in globally coupled oscillators with mean-field forcing.

Findings

Complex bifurcation structure with $$-rotational symmetry.

Resembles a 2:1 resonance tongue with inside 1:1 entrainment.

Diverse cluster solutions and stability regimes identified.

Abstract

We study two-cluster solutions of an ensemble of generic limit-cycle oscillators in the vicinity of a Hopf bifurcation, i.e. Stuart-Landau oscillators, with a nonlinear global coupling. This coupling leads to conserved mean-field oscillations acting back on the individual oscillators as a forcing. A reduction to two effective equations makes a linear stability analysis of the cluster solutions possible. These equations exhibit a $\pi$-rotational symmetry, leading to a complex bifurcation structure and a wide variety of solutions. In fact, the principal bifurcation structure resembles that of a 2:1 resonance tongue, while inside the tongue we observe an 1:1 entrainment.