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

• Published in: Nonlinearity Vol. 25; no. 5; pp. 1247 - 1273
• Main Authors: Giacomin, Giambattista, Pakdaman, Khashayar, Pellegrin, Xavier
• Format: Journal Article
• Language: English
• Published: Bristol Institute of Physics 01.05.2012; IOP Publishing
• Subjects: Asymptotic properties; Dynamic tests; Exact sciences and technology; Global analysis, analysis on manifolds; Invariants; Joining; Mathematical analysis; Mathematical methods in physics; Mathematics; Nonlinear dynamics; Nonlinearity; Ordinary differential equations; Oscillators; Other topics in mathematical methods in physics; Physics; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Strength; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Saddle point method; Zero; Coupled oscillator; Critical value; Nonlinear analysis; Long term variation; Equilibrium; Long term; Connecting; Saddle point; Global attractor; Equilibrium point; Non linear stability; White noise; Two-dimensional calculations; Trajectory; Dynamic model; Mathematical physics; Invariant curve
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/25/5/1247

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.