# Random Switching near Bifurcations

## TL;DR

This paper classifies stochastic switching processes near bifurcations, analyzing invariant measures and blow-up phenomena, with applications to nonlinear models.

## Contribution

It provides a comprehensive classification of piecewise deterministic Markov processes near various bifurcations and studies their invariant measures and blow-up conditions.

## Key findings

- Invariant measures exist for different switching rates
- Conditions for uniqueness and multiplicity of measures
- Identification of blow-up scenarios in models

## Abstract

The interplay between bifurcations and random switching processes of vector fields is studied. More precisely, we provide a classification of piecewise deterministic Markov processes arising from stochastic switching dynamics near fold, Hopf, transcritical and pitchfork bifurcations. We prove the existence of invariant measures for different switching rates. We also study, when the invariant measures are unique, when multiple measures occur, when measures have smooth densities, and under which conditions finite-time blow-up occurs. We demonstrate the applicability of our results for three nonlinear models arising in applications.