Limit cycles of a class of Liénard systems with restoring forces of seventh degree

The study of limit cycles for Liénard system is very important not only in theoretical studies but also in applications. In this paper, we study the number of limit cycles for a class of Liénard systems with restoring forces of seventh degree. Let H(n, m) denote the maximum number of limit cycles bifurcated from the generalized Liénard system x˙=y,y˙=−g(x)−f(x)y, where f(x) and g(x) are polynomials in x and degf=n,detg=m. We greatly improve the existing results of H(n, m) for m=7,n=4 and m=7,n=2n¯ with 4≤n¯≤20.