Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

Abstract

Spontaneous synchronization is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronization. The model comprises limit-cycle oscillators with distributed natural frequencies interacting through a mean-field coupling. Although more than forty years have passed since its introduction, the model continues to occupy the centre-stage of research in the field of non-linear dynamics, and is also widely applied to model diverse physical situations. In this brief review, starting with a derivation of the Kuramoto model and the synchronization phenomenon it exhibits, we summarize recent results on the study of a generalized Kuramoto model that includes inertial effects and stochastic noise. We describe the dynamics of the generalized model from a different yet a rather useful perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. A system is said to be long-range interacting if the inter-particle potential decays slowly as a function of distance. Using tools of statistical physics, we highlight the equilibrium and nonequilibrium aspects of the dynamics of the generalized Kuramoto model, and uncover a rather rich and complex phase diagram that it exhibits, which underlines the basic theme of intriguing emergent phenomena that are exhibited by many-body complex systems.

Analytical approach to synchronous states of globally coupled noisy rotators

“…The Kuramoto model of globally coupled phase oscillators [1] is a paradigmatic model to study synchronization phenomena [2][3][4][5]. In the thermodynamic limit, stationary (in a proper rotating reference frame) synchronized states can be found analytically for an arbitrary distribution of natural frequencies [6].…”

Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results

“…The Kuramoto model enjoys a unique status in the field of nonlinear dynamics [1,2,3,4,5,6,7]. It provides arguably the minimal framework to model the phenomenon of spontaneous synchronization commonly observed in nature [8,9], for example, among groups of fireflies flashing on and off in unison [10], in cardiac pacemaker cells [11], in electrochemical [12] and electronic [13] oscillators, in Josephson junction arrays [14] and in electrical power-grid networks [15], to name a few.…”

The Kuramoto–Lohe model and collective absorption of a photon

“…The statistical time evolution of oscillator phases in a network of coupled oscillators has been widely studied using the Kuramoto model [15][16][17][18][19][20][21][22]. The Kuramoto model is a simple statistical model, but appropriate for predicting synchronization thresholds for a remarkable variety of systems that exhibit some kind of periodicity.…”