Randomly switched vector fields sharing a zero on a common invariant face

TL;DR

This paper studies a piecewise deterministic Markov process with switching vector fields sharing a zero on a face, analyzing the impact of Lyapunov exponents of opposite signs, with applications to Lorenz systems and epidemiological models.

Contribution

It extends the analysis of switching vector fields to cases with Lyapunov exponents of opposite signs, especially when the process leaves an invariant face.

Findings

Behavior determined by Lyapunov exponents of opposite signs

Results apply to Lorenz vector fields with switching

Insights into SIRS models in random environments

Abstract

We consider a Piecewise Deterministic Markov Process given by random switching between finitely many vector fields vanishing at $0$. It has been shown recently that the behaviour of this process is mainly determined by the signs of Lyapunov exponents. However, results have only been given when all these exponents have the same sign. In this note, we consider the degenerate case where the process leaves invariant some face and results are stated when the Lyapunov exponents are of opposite signs. Applications are given to Lorenz vector fields with switching, and to SIRS model in random environment.