Full synchronization in a Kuramoto mean field game

Abstract

We consider the Kuramoto mean field game system which has been recently introduced and studied by Carmona, Cormier and Soner (2023). We review the existing literature on the subject and provide the asymptotics of the ergodic equilibria in the strong coupling limit, that is, when the parameter measuring the strength of interaction goes to infinity. We show that phase locking arises, that is, all the oscillators share the same phases. The argument is based on classical results about vanishing viscosity limits for viscosity solutions and on the properties of ergodic equilibria of the Kuramoto mean field game described by Cesaroni and Cirant (2024).