Limit Cycle Bifurcation Of A Class Of Liénard System And A Class Of Near-Hamiltonian System

In this paper,we mainly discuss a compound loop bifurcation and a heteroclinic bifurcation for a kind of Lienard system and a kind of near-Hamiltonian system.In the first chapter,we introduce the background of the research topic,the research status,and the main issues and related conclusions to be discussed in this article.In the second chapter,we mainly study the number of limit cycles for a kind of Lienard system x= y,y=-g(x)+?(?)aixiy under polynomial perturbations,where deg g(x)=5,and give new estimations for the lower bounds of the Hilbert number H(n,5)(10?n?20).In the third chapter,we mainly study the number of limit cycles near a heteroclinic loop and a compound loop with two nilpotent cuspidal saddles of order m,and give the expansion of the Melnikov function near the heteroclinic loop and the compound loop.The calculation formulas of the first several coefficients in the expansion are given.By the expansion of the Melnikov function near the heteroclinic loop,we further give the conditions for the near-Hamiltonian system to obtain limit cycles near the heteroclinic loop.When m=2,for both general near-Hamiltonian system and centrally symmetric near-Hamiltonian system we give the conditions to obtain limit cycles near the compound loop.Finally,by using the theorem given in this paper,we study the number of limit cycles for a concrete centrally symmetric near-Hamiltonian system.