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Stability and bifurcation for the Kuramoto model

Helge Dietert

TL;DR

The paper analyzes the mean-field limit of the Kuramoto model of globally coupled oscillators, establishing global stability for small coupling through Fourier-space energy methods and identifying a critical coupling $K_{ec}$ that governs damping of the order parameter, with a Landau-damping–like mechanism. It develops a comprehensive framework combining Volterra equations for the order parameter, linear stability via a Paley–Wiener/Gel’fand analysis, and nonlinear stability in exponential and algebraic norms, extending to a center-unstable manifold reduction when linear stability fails. This leads to a rigorous bifurcation analysis of partially locked states, including a detailed center-unstable reduction and concrete results for Gaussian (and related) velocity distributions, consistent with known bifurcation structures. The approach yields both global stability results and precise nonlinear dynamics near the incoherent state, providing a robust toolset for understanding synchronization and its bifurcations in the Kuramoto system and highlighting a deep connection to Landau damping in kinetic equations. The findings have significance for predicting and characterizing synchronization transitions in large ensembles of coupled oscillators, with potential extensions to noisy settings and broader mean-field models.

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.

Stability and bifurcation for the Kuramoto model

TL;DR

The paper analyzes the mean-field limit of the Kuramoto model of globally coupled oscillators, establishing global stability for small coupling through Fourier-space energy methods and identifying a critical coupling that governs damping of the order parameter, with a Landau-damping–like mechanism. It develops a comprehensive framework combining Volterra equations for the order parameter, linear stability via a Paley–Wiener/Gel’fand analysis, and nonlinear stability in exponential and algebraic norms, extending to a center-unstable manifold reduction when linear stability fails. This leads to a rigorous bifurcation analysis of partially locked states, including a detailed center-unstable reduction and concrete results for Gaussian (and related) velocity distributions, consistent with known bifurcation structures. The approach yields both global stability results and precise nonlinear dynamics near the incoherent state, providing a robust toolset for understanding synchronization and its bifurcations in the Kuramoto system and highlighting a deep connection to Landau damping in kinetic equations. The findings have significance for predicting and characterizing synchronization transitions in large ensembles of coupled oscillators, with potential extensions to noisy settings and broader mean-field models.

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 for . 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.

Paper Structure

This paper contains 12 sections, 36 theorems, 201 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Strategy and detailed results
  4. Acknowledgements
  5. Mean-field limit
  6. Global stability by energy method
  7. Stability of the linearised evolution
  8. Nonlinear stability
  9. Linear eigenmodes and spectral decomposition
  10. Center-unstable manifold reduction
  11. Bifurcation analysis
  12. Boundedness and convergence in the exponential norms

Key Result

Theorem 1

For a velocity distribution $g \in \mathcal{M}(\mathbb{R})$ suppose $\int_0^\infty |\hat{g}(\xi)| \mathrm{d} \xi < \infty$ and let If the coupling constant $K$ satisfies $K < K_{ec}$, then there exists a finite constant $c > 0$ and a bounded increasing weight $\phi \in C^1(\mathbb{R}^+)$ with $\phi(0) = 1$ so that for a solution to the Kuramoto equation with velocity marginal $g$ the energy funct

Figures (2)

  • Figure 1: The black line shows $\mathcal{L} \hat{g}(i \mathbb{R})$ with the enclosed green area $\mathcal{L} \hat{g}(\{z : \Re z > 0\})$ for the Gaussian distribution $g(\omega) = (2\pi)^{-1/2} e^{-\omega^2/2}$.
  • Figure 2: The black line shows $\mathcal{L} \hat{g}(i \mathbb{R})$ with the enclosed green area $\mathcal{L} \hat{g}(\{z : \Re z > 0\})$ for two Gaussian distributions with variance $1$ and centred at $\omega = \pm 1.5$.

Theorems & Definitions (74)

  • Remark
  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Definition 8
  • Lemma 9
  • ...and 64 more