On limit cycles of the Liénard equation with Z2 symmetry

This paper considers the limit cycles in the Liénard equation, described by x¨+f(x)x˙+g(x)=0, with Z2 symmetry (i.e., the vector filed is symmetric with the y-axis). Particular attention is given to the existence of small-amplitude (local) limit cycles around fine focus points when g(x) is a third-degree, odd polynomial function and f(x) is an even function. Such a system has three fixed points on the x-axis, with one saddle point at the origin and two linear centres which are symmetric with the origin. Based on normal form computation, it is shown that such a system can generate more limit cycles than the existing results for which only the origin is considered. In general, such a Liénard equation can have 2m small limit cycles, i.e., H(2m, 3) ⩾ 2m, where H denotes the Hilbert number of the system, 2m and 3 are the degrees of f and g, respectively.