Abstract: Building on Winfree’s work, the Kuramoto model (1975) has become the corner stone of mathematical models of collective synchronization, and has received attention in all natural sciences, engineering, and mathematics. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long term behavior of the system.  While the classical model postulates the dynamics of each oscillator in the form of a system of nonlinear ordinary differential equations, Yin, Mehta, Meyn, & Shanbhag (2010) use the mean-field game (MFG) formalism of Lasry & Lions, and Huang, Caines, & Malhame. In this talk, in addition to the Yin et.al model, we also introduce a simpler two state model which can be seen as a discretization of the original one.  We outline results showing that the mean field approach also delivers same type of results including the phase transition from incoherence to synchronization.  In particular, in the discrete setting we provide a comprehensive characterization of stationary and dynamic equilibria along with their stability properties. In all models, while the system is unsynchronized when the coupling is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to a full synchronization above a critical value of the interaction parameter. In the subcritical regime, the uniform distribution representing incoherence is the only stationary equilibrium. Above the critical interaction threshold, the uniform equilibrium becomes unstable and there is a multiplicity of stationary equilibria that are self-organizing. The discrete model with discounted cost present dynamic equilibria that spiral around the uniform distribution before converging to the self-organizing equilibria. With an ergodic cost, however, unexpected periodic equilibria around the uniform distribution emerge.

This talk is based on joint works with Rene Carmona and Felix Hoefer of Princeton, and Quentin Cormier of INRIA.