On the independent perturbation parameters and the number of limit cycles of a type of Liénard system

In this paper, we study a type of polynomial Liénard system of degree m(m≥2) with polynomial perturbations of degree n. We prove that the first order Melnikov function of such system has at most n+1−[n+1m+1] independent perturbation parameters which can be used to simplify this kind of systems. As an application, we study a type of Lienard systems for m=4,n=19,28 and obtain the new lower bounds of the maximal number of limit cycles.