Relaxation Oscillations and the Entry-Exit Function in Multi-Dimensional Slow-Fast Systems

TL;DR

This paper develops criteria for the existence and stability of relaxation oscillations in multi-dimensional slow-fast systems, generalizing the entry-exit relation, and applies these to ecological predator-prey models with rapid evolution.

Contribution

It introduces a generalized entry-exit relation for systems with multi-dimensional fast variables and derives conditions for relaxation oscillations, extending classical theory.

Findings

Existence of relaxation oscillations in multi-dimensional slow-fast systems.

Application of criteria to ecological predator-prey models.

Demonstration of oscillations due to stability changes at turning points.

Abstract

For a slow-fast system of the form $\dot{p}=\epsilon f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in \mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular closed orbit formed by trajectories of the limiting slow and fast systems. Assuming that the stability of $M_i$ changes along the slow trajectories at certain turning points, we derive criteria for the existence and stability of relaxation oscillations for the slow-fast system. Our approach is based on a generalization of the entry-exit relation to systems with multi-dimensional fast variables. We then apply our criteria to several predator-prey systems with rapid ecological evolutionary dynamics to show the existence of relaxation oscillations in these models.