Perturbation of a piecewise regular-singular Liénard system

Abstract

In this paper we discuss the perturbation of a generalized Liénard system, defined in two regions. For x≤0 the system is regular, while for x > 0 the system is singularly perturbed. An essential difference with other papers is that a degenerate canard point appears which requires a blow-up analysis.We prove that under generic conditions the perturbed system has at most one small-amplitude limit cycle and at most N canard limit cycles, where N is the number of local extrema of the function F(x) in the singular part of the Liénard system. For critical values of the control parameters the mechanism of a canard explosion is discussed.As an application we prove that the upper bound for the number of limit cycles in a Gause predator-prey system with a general monotonic functional response function with cut-off is two times the number of maxima of the natural prey growth function after the cut-off.