Local and global bifurcations to limit cycles in a class of Lienard equation

In this paper, we study limit cycles in the Lienard equation: x+f(x)x+g(x)=0 where f(x) is an even polynomial function with degree 2m, while g( x) is a third-degree, odd polynomial function. In phase space, the system has three fixed points, one saddle point at the origin and two linear centers which are symmetric about the origin. It is shown that the system can have 2m small ( local) limit cycles in the vicinity of two focus points and several large ( global) limit cycles enclosing all the small limit cycles. The method of normal forms is employed to prove the existence of the small limit cycles and numerical simulation is used to show the existence of large limit cycles.