Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise

TL;DR

This paper derives a finite-dimensional description for ensembles of identical phase oscillators under Cauchy noise, revealing exact stationary synchronous regimes and connecting to the Ott-Antonsen reduction.

Contribution

It introduces a method to exactly describe stationary regimes of oscillators with multiple harmonic forcing using finite complex order parameters.

Findings

Stationary regimes are described by wrapped Cauchy distributions.

Finite-dimensional equations are derived for systems with multiple harmonics.

Connection to Ott-Antonsen reduction is established for sinusoidal forcing.

Abstract

We study an ensembles of globally coupled or forced identical phase oscillators subject to independent white Cauchy noise. We demonstrate, that if the oscillators are forced in several harmonics, stationary synchronous regimes can be exactly described with a finite number of complex order parameters. The corresponding distribution of phases is a product of wrapped Cauchy distributions. For sinusoidal forcing, the Ott-Antonsen low-dimensional reduction is recovered.