Synchronization in a Kuramoto Mean Field Game

03/22/2023 12:30 pm - 1:30 pm
CMSA Room G10
Address: CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Mete Soner (Princeton University)

Title: Synchronization in a Kuramoto Mean Field Game

Abstract:  Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a  key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long term behavior of the system. While the system is not synchronized when this term is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to a full synchronization above a critical value of the interaction parameter.  We explore this system in the mean field formalism.  We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles determine endogenously their behaviors by minimizing a cost functional and eventually, settling in a Nash equilibrium.  The mean field game also exhibits a bifurcation from incoherence to self-organization.  This approach has found interesting applications including circadian rhythms and jet-lag recovery.  This is joint work with Rene Carmona of Princeton and Quentin Cormier of INRIA, Paris.