Dynamics of noisy oscillator populations beyond the Ott-Antonsen ansatz

TL;DR

This paper introduces a novel approach using circular cumulants to analyze large populations of noisy phase oscillators, extending the Ott-Antonsen framework to include noise effects and perturbations.

Contribution

It develops a cumulant-based method for describing noisy oscillator populations, providing a perturbation theory around the Ott-Antonsen solution.

Findings

Derived a closed system of equations for cumulants under noise.

Analyzed noise effects on Kuramoto and chimera states.

Extended Ott-Antonsen theory to noisy environments.

Abstract

We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, CHAOS 18, 037113 (2008)] and appear appropriate for constructing the perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulants, describing the dynamics of noisy ensembles, is derived. We exemplify the general theory by presenting the effect of noise on the Kuramoto system and on a chimera state in two symmetrically coupled populations.