Mathematical framework for breathing chimera states

TL;DR

This paper develops a semi-analytic mathematical framework to analyze breathing chimera states in nonlocally coupled oscillators, providing insights into their stability and macroscopic features.

Contribution

It introduces a novel approach based on integro-differential equations and self-consistency conditions to study nonstationary chimera states systematically.

Findings

Predicts macroscopic features of breathing chimeras

Derives stability conditions for breathing chimera solutions

Potentially applicable to other oscillator models using Ott-Antonsen reduction

Abstract

About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique.