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.
Highlights
Lienard equation is a classical and important nonlinear oscillator, modeling nonlinear vibrations in the damped, one-degree-of-freedom systems
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
We rigorously prove the existence of regular relaxation oscillations and canard relaxation oscillations under the specific parameter conditions
Summary
Lienard equation is a classical and important nonlinear oscillator, modeling nonlinear vibrations in the damped, one-degree-of-freedom systems. The limit cycles occurring in singularly perturbed Lienard systems are relaxation oscillations, characterized by the presence of phases in the cycle with different time scales: A phase of slow change is followed by a short phase of rapid change where the system jumps to the stage of slow variation In this direction, non-standard analysis was first introduced to detect canard relaxation oscillations in van der Pol equation [1]. We will extend the asymptotic analysis method of Eckhaus [10] to study the birth of relaxation oscillations in the following singularly perturbed generalized Lienard system,.
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