The Dynamics of a Kind of Liénard System with Sixth Degree and Its Limit Cycle Bifurcations Under Perturbations

Abstract

In this paper, the different topological types of phase portrait of the unperturbed Lienard system $$ {\dot{x}}=y,\ \ {\dot{y}}=-g(x)$$ are given, where $$\deg {g(x)}=6$$. We find that the expansion of the Melnikov function near any of closed orbits appeared in the above phase portraits, except a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle of order one, has been studied. In this paper, we give the expansion of the Melnikov function near this kind of heteroclinic loop. Further, we present the conditions to obtain limit cycles bifurcated from a compound loop with a hyperbolic saddle and a nilpotent saddle of order one, and apply it to study the number of limit cycles for a kind of Lienard system under perturbations.