Existence, stability and regularity of periodic solutions for nonlinear Fokker-Planck equations

TL;DR

This paper proves the existence, stability, and regularity of periodic solutions in nonlinear Fokker-Planck equations modeling mean-field interacting populations, using invariant manifold theory.

Contribution

It introduces a novel approach combining slow-fast analysis and invariant manifolds to establish periodic solutions and their smooth isochron maps in nonlinear PDEs.

Findings

Existence of stable periodic solutions for the nonlinear Fokker-Planck equations.

Construction of a smooth isochron map near the periodic solution.

Application of slow-fast and invariant manifold techniques to PDEs.

Abstract

We consider a class of nonlinear Fokker-Planck equations describing the dynamics of an infinite population of units within mean-field interaction. Relying on a slow-fast viewpoint and on the theory of approximately invariant manifolds we obtain the existence of a stable periodic solution for the PDE, consisting of probability measures. Moreover we establish the existence of a smooth isochron map in the neighborhood of this periodic solution.