On Bifurcation Delay: An Alternative Approach Using Geometric Singular Perturbation Theory

TL;DR

This paper uses geometric singular perturbation theory and the Exchange Lemma to analyze bifurcation delay, demonstrating that the maximum of the solution is exponentially small and establishing smoothness of the return map.

Contribution

It introduces an alternative approach to bifurcation delay analysis using geometric singular perturbation theory and extends the dimension to handle degeneracies.

Findings

Maximum of $z$ is of order $ ext{exp}(-1/ ext{epsilon})$

Return map is smooth up to arbitrary finite order in $ ext{epsilon}$

Provides a new perspective on bifurcation delay phenomena

Abstract

To explain the phenomenon of bifurcation delay, which occurs in planar systems of the form $\dot{x}=\epsilon f(x,z,\epsilon)$, $\dot{z}=g(x,z,\epsilon)z$, where $f(x,0,0)>0$ and $g(x,0,0)$ changes sign at least once on the $x$-axis, we use the Exchange Lemma in Geometric Singular Perturbation Theory to track the limiting behavior of the solutions. Using the trick of extending dimension to overcome the degeneracy at the turning point, we show that the limiting attracting and repulsion points are given by the well-known entry-exit function, and the maximum of $z$ on the trajectory is of order $\exp(-1/\epsilon)$. Also we prove smoothness the return map up to arbitrary finite order in $\epsilon$.