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

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.

Keywords:

• Dynamic stability
• mean field games
• Kuramoto model
• synchronization

2020 Mathematics Subject Classification:

• 35Q89
• 49N80
• 92B25

Acknowledgments

The authors are members of GNAMPA-INdAM.

Additional information

Funding

They were partially supported by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 “Mean-Field Games: models, theory, and computational aspects.”