Dynamical aspects of mean field plane rotators and the Kuramoto model

Abstract

The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N to infinity limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N to infinity limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K_c. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K_c.

18 pages, 1 figure

Cited by in corpus (28)

• Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus
• Propagation of chaos: a review of models, methods and applications. II. Applications
• Approximate solution to the stochastic Kuramoto model
• Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators
• A note on dynamical models on random graphs and Fokker-Planck equations
• Traveling Bands, Clouds, and Vortices of Chiral Active Matter
• Bounds for Turánians of modified Bessel functions
• On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
• Collective phase dynamics of globally coupled oscillators: Noise-induced anti-phase synchronization
• Ott-Antonsen ansatz is the only admissible truncation of a circular cumulant series
• Quenched limits and fluctuations of the empirical measure for plane rotators in random media
• Weakly interacting oscillators on dense random graphs
• Phase fluctuations in the ABC model
• Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions
• Higher order regularity of nonlinear Fokker-Planck PDEs with respect to the measure component
• Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels
• Distributed synaptic weights in a LIF neural network and learning rules
• Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations
• Effects of local fields in a dissipative Curie-Weiss model: Bautin bifurcation and large self-sustained oscillations
• Coexistence of stable limit cycles in a generalized Curie-Weiss model with dissipation
• Uniform in time weak propagation of chaos on the torus
• Propagation of chaos: a review of models, methods and applications. I. Models and methods
• Synchronization and spin-flop transitions for a mean-field XY model in random field
• Two-community noisy Kuramoto model with general interaction strengths: Part II
• Two-community noisy Kuramoto model with general interaction strengths: Part I
• Collective in-plane magnetization in a 2D XY macrospin system within the framework of generalized Ott-Antonsen theory
• Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators
• Bounds for the asymptotic order parameter of the stochastic Kuramoto model