Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model

E. Luçon and C. Poquet

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Abstract

The aim of the paper is to address the long time behavior of the Kuramoto model of mean-field coupled phase rotators, subject to white noise and quenched frequencies. We analyse the influence of the fluctuations of both thermal noise and frequencies (seen as a disorder) on a large but finite population of $N$ rotators, in the case where the law of the disorder is symmetric. On a finite time scale $[0,T]$, the system is known to be self-averaging: the empirical measure of the system converges as $N\to\infty$ to the deterministic solution of a nonlinear Fokker–Planck equation which exhibits a stable manifold of synchronized stationary profiles for large interaction. On longer time scales, competition between the finite-size effects of the noise and disorder makes the system deviate from this mean-field behavior. In the main result of the paper we show that on a time scale of order $\sqrt{N}$ the fluctuations of the disorder prevail over the fluctuations of the noise: we establish the existence of disorder-induced traveling waves for the empirical measure along the stationary manifold. This result is proved for fixed realizations of the disorder and emphasis is put on the influence of the asymmetry of these quenched frequencies on the direction and speed of rotation of the system. Asymptotics on the drift are provided in the limit of small disorder.

Résumé

Le but de ce travail est d’étudier le comportement en temps long du modèle de Kuramoto, défini par un système de rotateurs en interaction de type champ-moyen, perturbé par un bruit blanc et possédant des fréquences aléatoires gelées. Nous analysons l’influence des fluctuations induites par le bruit et les fréquences (vues comme un désordre pour le modèle) sur une population de $N$ rotateurs ($N$ grand mais fini), dans le cas où la loi du désordre est symétrique. Sur un intervalle de temps borné $[0,T]$, le système est auto-moyennant: la mesure empirique du système converge pour $N\to\infty$ vers la solution déterministe d’une équation de Fokker–Planck non linéaire possédant une variété stable de solutions stationnaires synchronisées pour une interaction suffisamment grande. Sur une échelle de temps plus grande, les effets de taille finie dûs à la présence du bruit et du désordre induisent une déviation macroscopique du système par rapport à ce comportement de champ-moyen. Le résultat principal de cet article montre que, sur une échelle de temps d’ordre $\sqrt{N}$, les fluctuations induites par le désordre l’emportent sur celles données par le bruit: nous montrons que le désordre induit l’existence de fronts pour la dynamique de la mesure empirique se propageant le long de la variété stationnaire. Ce résultat est valide pour une réalisation gelée du désordre. L’accent est mis sur l’influence de l’asymétrie des fréquences sur la direction et la vitesse de propagation du front et nous donnons une asymptotique de cette vitesse dans la limite de faible désordre.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1196-1240.

Dates
Received: 23 June 2015
Revised: 9 February 2016
Accepted: 16 March 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624040

Digital Object Identifier
doi:10.1214/16-AIHP753

Mathematical Reviews number (MathSciNet)
MR3689966

Zentralblatt MATH identifier
06775437

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 37N25: Dynamical systems in biology [See mainly 92-XX, but also 91-XX] 82C26: Dynamic and nonequilibrium phase transitions (general) 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10] 82C44: Dynamics of disordered systems (random Ising systems, etc.) 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]

Keywords
Kuramoto synchronization model Mean-field particle systems Disordered models Nonlinear Fokker–Planck PDE Long time dynamics Traveling waves Stochastic partial differential equations

Citation

Luçon, E.; Poquet, C. Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1196--1240. doi:10.1214/16-AIHP753. https://projecteuclid.org/euclid.aihp/1500624040


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