Synchronization for the Rough Kuramoto Model

TL;DR

This paper investigates how rough noise influences phase synchronization in the Kuramoto model, proving exponential convergence and analyzing long-term behavior using Lyapunov functions.

Contribution

It introduces a novel analysis of the Kuramoto model under rough noise, providing explicit convergence rates and basin size quantification.

Findings

Proves exponential convergence towards synchronization.

Quantifies the size of the random basin of attraction.

Shows long-term behavior depends on phases' mean evolution.

Abstract

We study the local synchronization of phases and frequencies for the Kuramoto model driven by rough noise. In particular, we prove exponential convergence towards synchronization and we give the explicit rate of convergence and quantify the size of the random basin of attraction. Furthermore, we show that the long time behavior of the system is determined by the evolution of phases' mean. Our result relies on the use of a Lyapunov function, capable of overriding the particular structure of the noise, taking in account only its intensity. Finally, we illustrate our analytical results and possible extensions with the help of numerical simulations.