The entry-exit function and geometric singular perturbation theory

TL;DR

This paper explores the entry-exit phenomenon in singular perturbation systems using geometric singular perturbation theory, clarifying the behavior of solutions near the critical manifold and analyzing the smoothness of return maps.

Contribution

It demonstrates how the linear case reduces to the quadratic case and applies geometric singular perturbation theory to explain entry-exit dynamics.

Findings

Solutions approach the x-axis for x<0 and are repelled for x>0.

The linear case can be reduced to the quadratic case.

The smoothness of the return map is discussed in the limit as epsilon approaches zero.

Abstract

For small $\epsilon>0$, the system $\dot x = \epsilon$, $\dot z = h(x,z,\epsilon)z$, with $h(x,0,0)<0$ for $x<0$ and $h(x,0,0)>0$ for $x>0$, admits solutions that approach the $x$-axis while $x<0$ and are repelled from it when $x>0$. The limiting attraction and repulsion points are given by the well-known entry-exit function. For $h(x,z,\epsilon)z$ replaced by $h(x,z,\epsilon)z^2$, we explain this phenomenon using geometric singular perturbation theory. We also show that the linear case can be reduced to the quadratic case, and we discuss the smoothness of the return map to the line $z=z_0$, $z_0>0$, in the limit $\epsilon\to0$.