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Relaxation Oscillations in Singularly Perturbed Generalized Lienard Systems with Non-Generic Turning Points

    Huan Hu Affiliation
    ; Jianhe Shen Affiliation
    ; Zheyan Zhou Affiliation
    ; Zhonghui Ou Affiliation

Abstract

Based on the asymptotic analysis technique developed by Eckhaus [Lecture Notes in Math., vol. 985, pp 449-494. Springer, Berlin, 1983], this paper aims to study the existence and the asymptotic behaviors of relaxation oscillations of regular and canard types in a singularly perturbed generalized Lionard system with a non-generic turning point. The singularly perturbed Lionard system considered in this paper is very general and numerous real world models like some biological ones can be rewritten in the form of this system after a series of transformations. Under certain conditions, we rigorously prove the existence of regular relaxation oscillations and canard relaxation oscillations under the specific parameter conditions. As an application, two biological models, namely, a FitzHugh-Nagumo model and a twodimensional predator-prey model with Holling-II response are studied, in which, the existence of regular relaxation oscillations and canard relaxation oscillations as well as the bifurcation curves are obtained.

Keyword : singular perturbation, relaxation oscillation, canard, non-generic turning point, generalized Lienard system

How to Cite
Hu, H., Shen, J., Zhou, Z., & Ou, Z. (2017). Relaxation Oscillations in Singularly Perturbed Generalized Lienard Systems with Non-Generic Turning Points. Mathematical Modelling and Analysis, 22(3), 389-407. https://doi.org/10.3846/13926292.2017.1315344
Published in Issue
May 19, 2017
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