On bifurcation delay: An alternative approach using Geometric Singular Perturbation Theory

• Published in: Journal of Differential Equations Vol. 262; no. 3; pp. 1617 - 1630
• Main Author: Hsu, Ting-Hao
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.02.2017
• Subjects: Bifurcation delay; Delay of instability; Entry-exit function; Exchange Lemma; Pontryagin delay; Slow–fast system; Bifurcation delay; Pontryagin delay; 34E20; 37C10; Slow–fast system; Exchange Lemma; Delay of instability; Entry-exit function
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2016.10.022

To explain the phenomenon of bifurcation delay, which occurs in planar systems of the form x˙=ϵf(x,z,ϵ), z˙=g(x,z,ϵ)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 minimum of z on the trajectory is of order exp⁡(−1/ϵ). Also we prove smoothness of the return map up to arbitrary finite order in ϵ.