Chaotic motions for a version of the Vlasov equation

TL;DR

This paper demonstrates that for a modified Vlasov equation with periodic potential and repulsive interactions, one can select initial particle speeds to induce chaotic orbits, highlighting the complex dynamics possible in such systems.

Contribution

It introduces a method to generate chaotic particle trajectories in a Vlasov system with periodic potential using Aubry-Mather theory.

Findings

Chaotic orbits can be achieved for any initial distribution.

Initial speeds can be chosen to produce chaos over infinite time.

The approach leverages Aubry-Mather theory and related ideas.

Abstract

We consider a version of the Vlasov equation on the circle under a periodic potential $V(x,t)$ and a repulsing smooth interaction $W$. We suppose that the Lagrangian for the single particle has chaotic orbits; using Aubry-Mather theory and ideas of W. Gangbo, A. Tudorascu and P. Bernard, we prove that, for any initial distribution of particles, it is possible to choose their initial speed in such a way to get a chaotic orbit on $[0,+\infty)$.