Stationary equilibria and their stability in a Kuramoto MFG with strong interaction

TL;DR

This paper analyzes stationary equilibria in a Kuramoto Mean Field Game with strong interactions, identifying two main types and examining their stability, thus advancing understanding of synchronization phenomena in large oscillator populations.

Contribution

It characterizes the stationary equilibria in the Kuramoto MFG and analyzes their stability, revealing only two possible equilibria under strong interaction conditions.

Findings

Only two stationary equilibria exist up to phase translation.

The self-organizing equilibrium can be locally stable.

Incoherent equilibrium is also characterized.

Abstract

Recently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of rational interacting oscillators. The MFG model exhibits several stationary equilibria, but the characterization of these equilibria and their ability to capture dynamic equilibria in long time remains largely open. In this paper, we demonstrate that, up to a phase translation, there are only two possible stationary equilibria: the incoherent equilibrium and the self-organizing equilibrium, given that the interaction parameter is sufficiently large. Furthermore, we present some local stability properties of the self-organizing equilibrium.