Discontinuous codimension-two bifurcation in a Vlasov equation

TL;DR

This paper investigates a unique discontinuous bifurcation in Vlasov equations caused by flat-top stationary states, revealing a codimension-two bifurcation through analytical and numerical methods.

Contribution

It identifies and analyzes a codimension-two bifurcation responsible for the discontinuous behavior in Vlasov equations with flat-top stationary states.

Findings

Discontinuous bifurcation occurs with flat-top stationary states.

The bifurcation is linked to a codimension-two bifurcation.

Analytical and numerical methods confirm the phenomenon.

Abstract

In a Vlasov equation, the destabilization of a homogeneous stationary state is typically described by a continuous bifurcation characterized by strong resonances between the unstable mode and the continuous spectrum. However, when the reference stationary state has a flat top, it is known that resonances drastically weaken, and the bifurcation becomes discontinuous. In this article, we use a combination of analytical tools and precise numerical simulations to demonstrate that this behavior is related to a codimension-two bifurcation, which we study in details.