Limit Cycle Bifurcations for a Kind of Liénard System with Degree Seven

Abstract

In this paper, we give the different topological types of phase portrait for Lienard system $\dot {x}=y, \dot {y}=-g(x)$ in the case that $\deg g(x)=7$ and the system have six and seven singular points, respectively. For its perturbed system, the expansion of the Melnikov function near any of the above closed orbits, except that the closed orbit is a compound loop passing through a nilpotent cusp and two hyperbolic saddles or passing through three hyperbolic saddles, has been studied. In this paper, as one of main results, for a near-Hamiltonian system, we give the expansion of the Melnikov function near a compound loop with a nilpotent cusp and two hyperbolic saddles. Based on this, we present the conditions to obtain limit cycles.