On the zeros of the Abelian integrals for a class of Lienard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this Letter, we investigated a class of Lienard systems of the form x-bar =y, y-bar =f(x)+yg(x) with degf=5 and degg=4. We proved that the related elliptic integrals of the Lienard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3