# Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems

## TL;DR

This paper classifies bifurcations of the critical set in singular fast-slow systems with one fast and one or two slow variables, linking these bifurcations to relaxation oscillations and their generic behaviors.

## Contribution

It provides a detailed classification of bifurcations of the critical set in systems with one fast and one slow variable, and conjectures about the case with two slow variables.

## Key findings

- Classification of bifurcations for one fast and one slow variable.
- Identification of bifurcations associated with loss of manifold structure.
- Conjecture on generic bifurcations for systems with two slow variables.

## Abstract

Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized by the slow variables. Using a distinguished parameter approach we are able to classify bifurcations for one fast and one slow variable. Some of these bifurcations are associated with the critical set losing manifold structure. We also conjecture a list of generic bifurcations of the critical set for one fast and two slow variables. We further consider how the bifurcations of the critical set can be associated with generic bifurcations of attracting relaxation oscillations under an appropriate singular notion of equivalence.