Exponential dephasing of oscillators in the Kinetic Kuramoto Model

TL;DR

This paper proves that in the kinetic Kuramoto model with weak coupling, the order parameter decays exponentially, leading oscillators to behave independently, akin to Landau damping in plasma physics.

Contribution

It demonstrates exponential dephasing in the kinetic Kuramoto model for small interactions, extending Landau damping concepts to coupled oscillators.

Findings

Exponential decay of the order parameter.

Asymptotic free flow behavior of oscillators.

Application of Landau damping techniques to the Kuramoto model.

Abstract

We study the kinetic Kuramoto model for coupled oscillators with coupling constant below the synchronization threshold. We manage to prove that, for any analytic initial datum, if the interaction is small enough, the order parameter of the model vanishes exponentially fast, and the solution is asymptotically described by a free flow. This behavior is similar to the phenomenon of Landau damping in plasma physics. In the proof we use a combination of techniques from Landau damping and from abstract Cauchy-Kowalewskaya theorem.