No abstract

In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

In this paper, we establish the existence of solitary wave in some KdV-mKdV equation with dissipative perturbation by applying geometric singular perturbation technique and Melnikov function. The distance of the stable manifold and unstable manifold are computed to show the existence of the homoclinic loop for the related ordinary differential equation systems on the slow manifold, which implies the existence of a solitary wave for the KdVmKdV equation with dissipative perturbation