Synchronization in a Kuramoto Mean Field Game

TL;DR

This paper explores synchronization phenomena in a mean field game version of the Kuramoto model, revealing phase transitions and stable equilibria through advanced mathematical techniques.

Contribution

It introduces a mean field game framework for the Kuramoto model, demonstrating phase transitions and synchronization phenomena with rigorous proofs.

Findings

Existence of phase transition at a critical interaction strength

Stable uniform distribution below critical value

Emergence of synchronized Nash equilibria above critical value

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 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.