Limit Cycles of a Class of Cubic Liénard Equations

In this paper, a class of polynomial Liénard systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ll}&\dot{x}=y-(a_{1}x+a_{2}x^{2}+a_{3}x^{3}),\\&\dot{y}=-(b_{1}x+b_{2}x^2+b_{3}x^3), \end{array}$$\end{document}is considered. Some conditions of the existence, non-existence and uniqueness of limit cycles are obtained by using Filippov transformations and Zhang’s theorem. We obtain that the above system has at most one limit cycle surrounding the origin if a1a3 < 0 or b2 = 0. And, one example is given to illustrate that the system can have three limit cycles.