Kuramoto model of synchronization: equilibrium and nonequilibrium aspects

By Shamik Gupta, A. Campa, Stefano Ruffo. 2014

202 citations recorded by the source corpus.

The phenomenon of spontaneous synchronization, particularly within the framework of the Kuramoto model, has been a subject of intense research over the years. The model comprises oscillators with distributed natural frequencies interacting through a mean-field coupling, and serves as a paradigm to study synchronization. In this review, we put forward a general framework in which we discuss in a unified way known results with more recent developments obtained for a generalized Kuramoto model that includes inertial effects and noise. We describe the model from a different perspective, highlighting the long-range nature of the interaction between the oscillators, and emphasizing the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view. In this review, we first introduce the model and discuss both for the noiseless and noisy dynamics and for unimodal frequency distributions the synchronization transition that occurs in the stationary state. We then introduce the generalized model, and analyze its dynamics using tools from statistical mechanics. In particular, we discuss its synchronization phase diagram for unimodal frequency distributions. Next, we describe deviations from the mean-field setting of the Kuramoto model. To this end, we consider the generalized Kuramoto dynamics on a one-dimensional periodic lattice on the sites of which the oscillators reside and interact with one another with a coupling that decays as an inverse power-law of their separation along the lattice. For two specific cases, namely, in the absence of noise and inertia, and in the case when the natural frequencies are the same for all the oscillators, we discuss how the long-time transition to synchrony is governed by the dynamics of the mean-field mode (zero Fourier mode) of the spatial distribution of the oscillator phases.

Subfields: Cognitive Neuroscience, Computer Networks and Communications, Global and Planetary Change

View Paper: https://www.semanticscholar.org/paper/ff11529aa29bb5af71832d485564eb74bb8e46d6

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Citations in the corpus

The Kuramoto model in complex networks (2015, background): “[314] provided a general meanfield analysis framework of the second-order Kuramoto model with noise, focusing on the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view.”
Synchronization in complex networks of phase oscillators: A survey (2014, background): “We also remark that a treatment of (17) as a stochastic differential equation (in the limit of zero additivewhite noise) results in a Fokker–Planck equation analogous to the continuum-limit model (Acebrón et al., 2005; Crawford, 1994; Gupta et al., 2014; Strogatz, 2000).”