Result: Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

Title:
Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators
Authors:
GIACOMIN, Giambattista, PAKDAMAN, Khashayar, PELLEGRIN, Xavier
Source:
Nonlinearity (Bristol. Print). 25(5):1247-1273
Publisher Information:
Bristol: Institute of Physics, 2012.
Publication Year:
2012
Physical Description:
print, 29 ref
Original Material:
INIST-CNRS
Subject Terms:
Mathematics, Mathématiques, Theoretical physics, Physique théorique, Sciences exactes et technologie, Exact sciences and technology, Sciences et techniques communes, Sciences and techniques of general use, Mathematiques, Mathematics, Analyse mathématique, Mathematical analysis, Equations différentielles, Ordinary differential equations, Topologie. Variétés et complexes cellulaires. Analyse globale et analyse sur variétés, Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds, Analyse globale, analyse sur des variétés, Global analysis, analysis on manifolds, Probabilités et statistiques, Probability and statistics, Théorie des probabilités et processus stochastiques, Probability theory and stochastic processes, Processus particuliers (théorie du renouvellement, processus de renouvellement markoviens, processus semi-markoviens, modèles de la mécanique statistique, applications diverses), Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications), Physique, Physics, Generalites, General, Méthodes mathématiques en physique, Mathematical methods in physics, Divers, Other topics in mathematical methods in physics, Analyse non linéaire, Nonlinear analysis, análisis no lineal, Attracteur global, Global attractor, Atractor global, Branchement, Connecting, Acometida, Bruit blanc, White noise, Ruido blanco, Calcul 2 dimensions, Two-dimensional calculations, Equilibre, Equilibrium, Equilibrio, Long terme, Long term, Largo plazo, Modèle dynamique, Dynamic model, Modelo dinámico, Méthode col, Saddle point method, Método punto en puerto, Oscillateur couplé, Coupled oscillator, Oscilador acoplamiento, Physique mathématique, Mathematical physics, Física matemática, Point col, Saddle point, Punto silla, Point équilibre, Equilibrium point, Punto equilibrio, Stabilité non linéaire, Non linear stability, Estabilidad no lineal, Trajectoire, Trajectory, Trayectoria, Valeur critique, Critical value, Valor crítico, Variation long terme, Long term variation, Variación largo plazo, Zéro, Zero, Cero, 34C15, 34D45, 35B42, 37B25, 60K40, Courbe invariante, Invariant curve, Ensemble normal, Existence globale, Hyperbolicité, Point selle, Unicité, Variété inertielle
Document Type:
Academic journal Article
File Description:
text
Language:
English
Author Affiliations:
Université Paris Diderot (Paris 7) and Laboratoire de Probabilités et Modèles Aléatoires (CNRS), UFR Mathématiques, Case 7012 (site Chevaleret, 75205 Paris, France
Institut Jacques Monod, CNRS, UMR 7592, Université Paris Diderot, Sorbonne Paris Cité, 750205 Paris, France
ISSN:
0951-7715
Access URL:
http://pascal-francis.inist.fr/vibad/index.php?action=search&terms=25877417
Rights:
Copyright 2015 INIST-CNRS
CC BY 4.0
Sauf mention contraire ci-dessus, le contenu de cette notice bibliographique peut être utilisé dans le cadre d’une licence CC BY 4.0 Inist-CNRS / Unless otherwise stated above, the content of this bibliographic record may be used under a CC BY 4.0 licence by Inist-CNRS / A menos que se haya señalado antes, el contenido de este registro bibliográfico puede ser utilizado al amparo de una licencia CC BY 4.0 Inist-CNRS
Notes:
Mathematics

Theoretical physics
Accession Number:
edscal.25877417
Database:
PASCAL Archive

Further Information

We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long-term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Otherwise, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disc composed of radial trajectories connecting a saddle-point equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and coherent (or synchronized) equilibria. We prove in particular nonlinear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero.