Landau damping to partially locked states in the Kuramoto model

TL;DR

This paper proves the spectral and local asymptotic stability of partially locked states in the Kuramoto model using Landau damping mechanisms, extending stability results to heterogeneous, singular measures without dissipation.

Contribution

It establishes the first explicit spectral stability criterion and proves local asymptotic stability of PLS in weak topology for a broad class of frequency distributions, beyond the Ott-Antonsen ansatz.

Findings

Spectral stability criterion for PLS derived

Local asymptotic stability proven in weak topology

Landau damping demonstrated for heterogeneous equilibria

Abstract

In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that incorporate the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology, or to specific invariant subspaces (obtained via the so-called Ott-Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. Besides, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here, we establish an explicit criterion for their spectral stability and we prove their local asymptotic stability in weak topology, for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We show in particular that no loss of generality results in assuming the OA ansatz. To our best knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria, in absence of dissipation.