Limit cycles of some polynomial Liénard systems

In this paper, we study the number of limit cycles of some polynomial Liénard systems. By the Melnikov functions and the methods of Hopf, homoclinic and heteroclinic bifurcation theory, we prove that H(2,5)⩾3, H(4,5)⩾5, H(6,5)⩾10, H(8,5)⩾10.