On the existence of oscillating solutions in non-monotone Mean-Field Games

TL;DR

This paper demonstrates the existence of multiple oscillating solutions in non-monotone Mean-Field Game systems, using bifurcation theory and numerical analysis to explore their properties and behaviors.

Contribution

It introduces the first rigorous proof of infinite oscillating solution branches in non-monotone Mean-Field Games, combining bifurcation analysis with numerical validation.

Findings

Existence of infinitely many non-trivial solution branches.

Solutions exhibit oscillatory behavior near trivial solutions.

Numerical models confirm theoretical bifurcation predictions.

Abstract

For non-monotone single and two-populations time-dependent Mean-Field Game systems we obtain the existence of an infinite number of branches of non-trivial solutions. These non-trivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutions predicted by bifurcation theory, and to study further properties of branches far away from bifurcation points.