Synchronization of Coupled Oscillators is a Game

The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, and economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population linear quadratic Gaussian (LQG) model of Huang et al. 2007, comprised here of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function. The states are interpreted as phase angles for a collection of heterogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents "opt out\´ of the game, setting their controls to zero, resulting in the "incoherence\´ equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the partial differential equation (PDE) model about the incoherence equilibrium. A critical value of the control cost parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.