Stationary and Traveling Wave States of the Kuramoto Model with an Arbitrary Distribution of Frequencies and Coupling Strengths

TL;DR

This paper develops a comprehensive theoretical framework for the Kuramoto model with arbitrary frequency and coupling distributions, deriving equations for wave states and stability conditions, applicable across various scientific fields.

Contribution

It introduces a general approach to analyze stationary and traveling wave states in the Kuramoto model with arbitrary distributions, extending previous results.

Findings

Derived equations for wave state parameters

Proposed empirical stability conditions

Unified previous theoretical results

Abstract

We consider the Kuramoto model of an ensemble of interacting oscillators allowing for an arbitrary distribution of frequencies and coupling strengths. We define a family of traveling wave states as stationary in a rotating frame, and derive general equations for their parameters. We suggest empirical stability conditions which, for the case of incoherence, become exact. In addition to making new theoretical predictions, we show that many earlier results follow naturally from our general framework. The results are applicable in scientific contexts ranging from physics to biology.