Heteroclinic Switching between Chimeras in a Ring of Six Oscillator Populations

TL;DR

This paper investigates heteroclinic switching between chimera states in a ring of six coupled oscillator populations, revealing how heterogeneity influences the stability and dynamics of these complex states.

Contribution

It introduces a detailed analysis of heteroclinic cycles and switching dynamics in a ring of oscillators, combining theoretical and finite-size system approaches.

Findings

Heteroclinic cycles support switching between chimera states.

Small heterogeneity leads to asymptotically attracting heteroclinic switching.

Large heterogeneity replaces heteroclinic orbits with stable chimera states.

Abstract

In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz, and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.