The entry-exit theorem and relaxation oscillations in slow-fast planar systems

• Published in: Journal of Differential Equations Vol. 268; no. 11; pp. 7220 - 7249
• Main Authors: Ai, Shangbing, Sadhu, Susmita
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.05.2020
• Subjects: Delay of loss of stability; Entry-exit function; Orbital stability; Relaxation oscillations; Uniqueness; 34D15; 34C26; Delay of loss of stability; Relaxation oscillations; Uniqueness; Orbital stability; 37C27; Entry-exit function; 34E05
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2019.11.067

The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. The general existing proofs of this theorem depend on Fenichel's geometric singular perturbation theory and blow-up techniques. In this work, we give a short and elementary proof of the entry-exit theorem based on a direct study of asymptotic formulas of the underlying solutions. We employ this theorem to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of relaxation oscillations. The results are then applied to a diffusive predator-prey model with Holling type II functional response to establish periodic traveling wave solutions. Furthermore, we extend our work to another class of slow-fast systems that can have multiple orbits exhibiting relaxation oscillations, and subsequently apply the results to a two time-scale Holling-Tanner predator-prey model with Holling type IV functional response. It is generally assumed in the literature that the non-trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations; we do not make this assumption in this paper.