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Landau damping to partially locked states in the Kuramoto model

Helge Dietert, Bastien Fernandez, David Gérard-Varet

TL;DR

This paper establishes a rigorous Landau-damping-type stabilization of partially locked states in the mean-field Kuramoto model by proving a sharp spectral stability criterion in a weak Fourier-based norm. The authors introduce a functional framework that accommodates singular measures and derives a 2×2 matrix condition det(I−(K/2)M(λ,r_s)) for eigenvalues, ensuring that 0 is the only nonnegative eigenvalue and is simple. Under this spectral condition, they prove nonlinear local asymptotic stability (modulo rotation) of the entire PLS circle, including heterogeneous and irregular equilibria, without dissipation. The work also clarifies when the Ott–Antonsen ansatz is without loss of generality and applies the theory to symmetric and bi-Cauchy marginals, highlighting bifurcation structures and stability of PLS branches. Overall, the results provide a first rigorous Landau-damping-type convergence to heterogeneous stationary states in a dissipation-free, mean-field setting, with implications for Vlasov-type dynamics and related coupled-oscillator models.

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.

Landau damping to partially locked states in the Kuramoto model

TL;DR

This paper establishes a rigorous Landau-damping-type stabilization of partially locked states in the mean-field Kuramoto model by proving a sharp spectral stability criterion in a weak Fourier-based norm. The authors introduce a functional framework that accommodates singular measures and derives a 2×2 matrix condition det(I−(K/2)M(λ,r_s)) for eigenvalues, ensuring that 0 is the only nonnegative eigenvalue and is simple. Under this spectral condition, they prove nonlinear local asymptotic stability (modulo rotation) of the entire PLS circle, including heterogeneous and irregular equilibria, without dissipation. The work also clarifies when the Ott–Antonsen ansatz is without loss of generality and applies the theory to symmetric and bi-Cauchy marginals, highlighting bifurcation structures and stability of PLS branches. Overall, the results provide a first rigorous Landau-damping-type convergence to heterogeneous stationary states in a dissipation-free, mean-field setting, with implications for Vlasov-type dynamics and related coupled-oscillator models.

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.

Paper Structure

This paper contains 20 sections, 15 theorems, 179 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Landau damping in the Kuramoto model
  3. The partially locked states
  4. Main result
  5. The Cauchy problem for the norm $\|\cdot\|_{a,0}$
  6. Spectral analysis of the linearized dynamics
  7. Resolvent estimates for $L_1$
  8. The spectrum of ${\cal L}$
  9. Derivation of the stability condition
  10. Proof of nonlinear stability
  11. Polar-type coordinates
  12. Proof of Theorem \ref{['thmNL']}
  13. Analysis of the forced linear equation
  14. Proof of Proposition \ref{['ASYMPPROJECT']}
  15. Stability condition: analysis and examples
  16. ...and 5 more sections

Key Result

Theorem 2.1

Assume that $g\in L^\infty(\mathbb{R})$, $\|\widehat{g}\|_a <+\infty$ for some $a>0$ and a PLS $f_\text{s}$ with marginal density $g$ and order parameter $r_\text{s}\in\mathbb{R}^+$ exists and satisfies the following conditions: Then, there exists $\epsilon,b>0$ such that for every probability measure $f_\text{\rm in}$ with marginal density $g$ and satisfying there exists $\Theta_\infty\in\mathb

Figures (1)

  • Figure 1: Schematic PLS bifurcation diagram for the bi-Cauchy distribution $g_{\Delta,\omega_0}$ with $\omega_0=2\Delta$ (obtained using the graph of the function $\Psi_{\Delta,\omega_0}$, see text).

Theorems & Definitions (16)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 5.1
  • Lemma 5.2
  • ...and 6 more