Limit Cycle Bifurcations Near A Cuspidal Loop For A General Near-Hamiltonian System

In the qualitative theory of the ordinary differential equations,one of the main problems is to study the number and distribution of the limit cycles for a near-Hamiltonian systems.To study limit cycle bifurcations,the first-order Melnikov function plays an important role.We suppose that the equation H(x,y)=?_c defines a cuspidal loop,we have obtained the expansions of the Melnikov functions near the cuspidal loop.How to compute more coefficients of the expansions of Melnikov functions near a cuspidal loop is very difficult.In this paper,we main study the limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system.We give a method to get more coefficients of the expansions of the Melnikov functions and find more limit cycles near the cuspidal loop.As an application example,we study a polynomial perturbation of degree 9 for a near-Hamiltonian system with a cuspidal loop and found 12 limit cycles.This paper includes three chapters:In chapter 1,we briefly introduce the research current situation,historical background,the method which we use and the main results of this paper.In chapter 2,we first introduce some preliminary.Then,we present the main theorems and the proof of our paper.In chapter 3,as an application example,we study a polynomial perturbation of degree 9 for a near-Hamiltonian system with a cuspidal loop and found 12 limit cycles.