Hyperbolic angular statistics for globally coupled phase oscillators

TL;DR

This paper introduces a hyperbolic angular statistical framework for globally coupled phase oscillators, extending the Kuramoto-Sakaguchi model with a multiplicative noise component, leading to analytically tractable synchronization conditions.

Contribution

It presents a novel hyperbolic drifted Brownian motion approach to generalize the Kuramoto-Sakaguchi dynamics with explicit control parameters.

Findings

Derivation of invariant measures as von Mises distributions

Analytical relation between control parameters at synchronization onset

Extension of Kuramoto-Sakaguchi model with multiplicative noise

Abstract

We analytically discuss a multiplicative noise generalization of the Kuramoto-Sakaguchi dynamics for an assembly of globally coupled phase oscillators. In the mean field limit, the resulting class of invariant measures coincides with a generalized, two parameter family of angular von Mises probability distributions which is governed by the exit law from the unit disc of a hyperbolic drifted Brownian motion. Our dynamics offers a simple yet analytically tractable generalization of Kuramoto-Sakaguchi dynamics with two control parameters. We derive an exact and very compact relation between the two control parameters at the onset of phase oscillators synchronization.