Stability and bifurcation for the Kuramoto model

TL;DR

This paper analyzes the stability and bifurcation phenomena in the Kuramoto model of coupled oscillators, revealing conditions for damping, exponential decay, and eigenmode behavior using Fourier analysis and manifold reduction.

Contribution

It introduces a rigorous Fourier space approach to establish global stability, damping, and bifurcation analysis for the Kuramoto model with regular velocity distributions.

Findings

Global stability result for the mean-field Kuramoto model

Identification of damping and exponential decay in stable regimes

Center-unstable manifold reduction for bifurcation analysis

Abstract

We study the mean-field limit of the Kuramoto model of globally coupled oscillators. By studying the evolution in Fourier space and understanding the domain of dependence, we show a global stability result. Moreover, we can identify function norms to show damping of the order parameter for velocity distributions and perturbations in $\mathcal{W}^{n,1}$ for $n > 1$. Finally, for sufficiently regular velocity distributions we can identify exponential decay in the stable case and otherwise identify finitely many eigenmodes. For these eigenmodes we can show a center-unstable manifold reduction, which gives a rigorous tool to obtain the bifurcation behaviour. The damping is similar to Landau damping for the Vlasov equation.