Singularities of invariant densities for random switching between two linear ODEs in 2D

TL;DR

This paper analyzes how the invariant density of a 2D system with two linear vector fields behaves, identifying singularities related to switching rates and contraction, with implications for biological models.

Contribution

It characterizes the singularities of invariant densities in a stochastic switching system and establishes boundedness properties, advancing understanding of such dynamical systems.

Findings

Identifies singularities of invariant densities based on switching and contraction rates

Proves the invariant density remains bounded away from singularities

Provides biological examples motivating the mathematical analysis

Abstract

We consider a planar dynamical system generated by two stable linear vector fields with distinct fixed points and random switching between them. We characterize singularities of the invariant density in terms of the switching rates and contraction rates. We prove boundedness away from those singularities. We also discuss some motivating biological examples.