Characterizing Slow Exit and Entrance Points

TL;DR

This paper enhances the analysis of slow exit and entrance points in dynamical systems by integrating Geometric Singular Perturbation Theory with Conley Index Theory, providing new insights and practical interpretations.

Contribution

It demonstrates how Fenichel Normal Form can simplify Conley index applications and bridges the gap between GSPT and Conley Theory with practical examples.

Findings

Fenichel Normal Form simplifies Conley index calculations.

Conley's conditions can be interpreted through averaging techniques.

Applications to van der Pol and Morris-Lecar models illustrate the methods.

Abstract

Geometric Singular Perturbation Theory (GSPT) and Conley Index Theory are two powerful techniques to analyze dynamical systems. Conley already realized that using his index is easier for singular perturbation problems. In this paper, we will revisit Conley's results and prove that the GSPT technique of Fenichel Normal Form can be used to simplify the application of Conley index techniques even further. We also hope that our results provide a better bridge between the different fields. Furthermore we show how to interpret Conley's conditions in terms of averaging. The result are illustrated by the two-dimensional van der Pol equation and by a three-dimensional Morris-Lecar model.