Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories

TL;DR

This paper presents an exact finite-dimensional reduction for populations of noisy oscillators, extending the Ott-Antonsen and Watanabe-Strogatz theories to describe transient dynamics beyond invariant subspaces.

Contribution

It introduces a novel exact reduction method for oscillator populations outside the Ott-Antonsen manifold, linking it to Watanabe-Strogatz theory for identical oscillators.

Findings

Reduces dynamics to three complex variables plus a constant function.

Connects the reduction to Watanabe-Strogatz equations for noise-free oscillators.

Enables analysis of transient dynamics in perturbed oscillator ensembles.

Abstract

Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities, or equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and thus describes the asymptotic dynamics. Here we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show, that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles.