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On the trend to global equilibrium for Kuramoto Oscillators

Javier Morales, David Poyato

Abstract

In this paper, we study the convergence to the stable equilibrium for Kuramoto oscillators. Specifically, we derive estimates on the rate of convergence to the global equilibrium for solutions of the Kuramoto-Sakaguchi equation in a large coupling strength regime from generic initial data. As a by-product, using the stability of the equation in the Wasserstein distance, we quantify the rate at which discrete Kuramoto oscillators concentrate around the global equilibrium. In doing this, we achieve a quantitative estimate in which the probability that the oscillators will concentrate at the given rate tends to one as the number of oscillators increases. Among the essential steps in our proof are: 1) An entropy production estimate inspired by the formal Riemannian structure of the space of probability measures, first introduced by F. Otto in [35]; 2) A new quantitative estimate on the instability of equilibria with antipodal oscillators based on the dynamics of norms of the solution in sets evolving by the continuity equation; 3) The use of generalized local logarithmic Sobolev and Talagrand type inequalities, similar to the ones derived by F. Otto and C. Villani in [36]; 4) The study of a system of coupled differential inequalities, by a treatment inspired by the work of L. Desvillettes and C. Villani [13]. Since the Kuramoto-Sakaguchi equation is not a gradient flow with respect to the Wasserstein distance, we derive such inequalities under a suitable fibered transportation distance.

On the trend to global equilibrium for Kuramoto Oscillators

Abstract

In this paper, we study the convergence to the stable equilibrium for Kuramoto oscillators. Specifically, we derive estimates on the rate of convergence to the global equilibrium for solutions of the Kuramoto-Sakaguchi equation in a large coupling strength regime from generic initial data. As a by-product, using the stability of the equation in the Wasserstein distance, we quantify the rate at which discrete Kuramoto oscillators concentrate around the global equilibrium. In doing this, we achieve a quantitative estimate in which the probability that the oscillators will concentrate at the given rate tends to one as the number of oscillators increases. Among the essential steps in our proof are: 1) An entropy production estimate inspired by the formal Riemannian structure of the space of probability measures, first introduced by F. Otto in [35]; 2) A new quantitative estimate on the instability of equilibria with antipodal oscillators based on the dynamics of norms of the solution in sets evolving by the continuity equation; 3) The use of generalized local logarithmic Sobolev and Talagrand type inequalities, similar to the ones derived by F. Otto and C. Villani in [36]; 4) The study of a system of coupled differential inequalities, by a treatment inspired by the work of L. Desvillettes and C. Villani [13]. Since the Kuramoto-Sakaguchi equation is not a gradient flow with respect to the Wasserstein distance, we derive such inequalities under a suitable fibered transportation distance.

Paper Structure

This paper contains 30 sections, 36 theorems, 428 equations.

Table of Contents

  1. Introduction
  2. The Kuramoto model
  3. The gradient flow structure and stationary solutions
  4. Statement of the problem and main results
  5. Ingredients
  6. Strategy
  7. Displacement concavity and entropy production
  8. Small dissipation regime and lower bounds in the order parameter
  9. Instability of the antipodal equilibria
  10. Sliding norms
  11. The system
  12. Local displacement convexity and Talagrand type inequalities
  13. Functional inequalities and a fibered Wasserstein distance
  14. A fibered Wasserstein distance and relation to dissipation
  15. Convergence and uniqueness of the global equilibria
  16. ...and 15 more sections

Key Result

Theorem \oldthetheorem

Let $f_{0}$ be contained in $C^{1}(\mathbb{T}\times\mathbb{R})$ and let $g$ be compactly supported in $[-W,W]$. Consider the unique global-in-time classical solution $f=f(t,\theta,\omega)$ to E-KS. Then, there exists a universal constant $C$ such that if then we can find a time $T_{0}$ with the property that and for every $t$ in $[T_{0},\infty)$. Here, is the unique global equilibrium of the Ku

Theorems & Definitions (75)

  • Remark 1.1
  • Theorem \oldthetheorem
  • Corollary 1.1
  • Definition 2.1
  • Lemma 2.1
  • Corollary 2.1
  • Theorem \oldthetheorem
  • proof
  • Corollary 3.1
  • proof
  • ...and 65 more