  The classical Kuramoto model is studied in the setting of an infinite horizon
mean field game. The system is shown to exhibit both synchronization and phase
transition. Incoherence below a critical value of the interaction parameter is
demonstrated by the stability of the uniform distribution. Above this value,
the game bifurcates and develops self-organizing time homogeneous Nash
equilibria. As interactions become stronger, these stationary solutions become
fully synchronized. Results are proved by an amalgam of techniques from
nonlinear partial differential equations, viscosity solutions, stochastic
optimal control and stochastic processes.
