GLOBAL DYNAMICS OF A POLYNOMIAL LIÉNARD SYSTEM

When studying traveling wave solutions of a generalized Burgers–Fisher equation we obtained a Liénard system of the form $ \frac{{\rm d}x}{{\rm d}\tau}=y, \ \frac{{\rm d}y}{{\rm d}\tau}=- x(x^{m}-1)-\varepsilon \left(a+ x^{m}\right)y $, where $ m\geq1 $ is an integer and $ \varepsilon $ is a small parameter. In this paper, we present a complete analysis for its global dynamics for all $ m\geq 1 $, giving necessary and sufficient conditions for the existence of one, two or three limit cycles and for the existence of homoclinic or double homoclinic loops by applying new methods established in this paper.