Synchronization in a Kuramoto mean field game
René Carmona, Quentin Cormier, H. Meté Soner
Abstract
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 get 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.
Topics & Concepts
Kuramoto modelMathematicsSynchronization (alternating current)Mean field theoryStability (learning theory)Applied mathematicsMathematical economicsViscosity solutionField (mathematics)Nonlinear systemStochastic differential equationInitial value problemStationary distributionStatistical physicsMathematical analysisPure mathematicsComputer scienceStatisticsCombinatoricsTopology (electrical circuits)PhysicsMarkov chainMachine learningQuantum mechanicsNonlinear Dynamics and Pattern FormationEcosystem dynamics and resilienceComplex Systems and Time Series Analysis