Dephasing of Kuramoto oscillators in kinetic regime towards a fixed asymptotically free state

TL;DR

This paper analyzes the kinetic Kuramoto model, demonstrating that small interactions lead to solutions approaching asymptotically free states with order parameters decaying over time, akin to Landau damping in plasma physics.

Contribution

It proves the existence of solutions near asymptotically free states with decay rates depending on regularity, extending understanding of dephasing in coupled oscillators.

Findings

Order parameter decays exponentially with analytical regularity.

Order parameter decays polynomially with Sobolev regularity.

Solutions approach asymptotically free states under small interactions.

Abstract

We study the kinetic Kuramoto model for coupled oscillators. We prove that for any regular asymptotically free state, if the interaction is small enough, it exists a solution which is asymptotically close to it. For this class of solution the order parameter vanishes to zero, showing a behavior similar to the phenomenon of Landau damping in plasma physics. We obtain an exponential decay of the order parameter in the case on analytical regularity of the asymptotic state, and a polynomial decay in the case of Sobolev regularity.