Dynamics of oscillator populations globally coupled with distributed phase shifts

TL;DR

This paper demonstrates that a large population of globally coupled oscillators with random phase shifts can be effectively described by a non-random system with a modified coupling function, simplifying analysis of complex oscillator networks.

Contribution

It introduces a reduction method that replaces random phase shifts with an effective coupling function, applicable to noisy and frequency-distributed oscillators, and discusses its limitations.

Findings

Effective coupling function is a convolution of original coupling and phase shift distribution.

Reduction holds for fully disordered and partially disordered cases under certain stability conditions.

The reduction may fail with complex coupling functions causing multistability.

Abstract

We consider a population of globally coupled oscillators in which phase shifts in the coupling are random. We show that in the maximally disordered case, where the pairwise shifts are i.i.d. random variables, the dynamics of a large population reduces to one without randomness in the shifts but with an effective coupling function, which is a convolution of the original coupling function with the distribution of the phase shifts. This result is valid for noisy oscillators and/or in presence of a distribution of natural frequencies. We argue also, using the property of global asymptotic stability, that this reduction is valid in a partially disordered case, where random phase shifts are attributed to the forced units only. However, the reduction to an effective coupling in the partially disordered noise-free situation may fail if the coupling function is complex enough to ensure the multistability of locked states.