Bifurcations Of Limit Cycles Near A Z₂-equivariant Compound Cycle In A Lienard System

As we know, the Hilberts16th problem is to considerthe least upper bound of the number of limit cycles and their relative locations in polynomial vector fields. More than one hundred years has passed since Hilbert presented the16th problem in1900. Although this problem has not beem completely solved for hundreds of years, Chinese and foreign mathematicians have done much research on this topic and established many kinds of theories, also obtained fruitful re suits. In this paper, we investigate the numberof limit cycles of a kind of Li\{e}nard polynomial system. This paper consists of five chapters, the particular contents of each c hap te rare as follows: In chapter1, our main purpose is to introduce the background ofour research and main topic s that we will study in the following chapters. At the same time, we give a description of our methods and re suits o b ta ine d in this chapter of this thesis. In chapter2, we expressed the main results and gave some introduction of the system we investigated. Erst, we investigated the properties of the system we are going to study, which will be useful in the following chapters, then we expressed the main re suits of ourpaper. In chapter3, we give some preliminary lemmas which will be useful in proving the main results. In this chapter, we give several important lemmas, which are important to investigate the polynomial differential system. Also, we have proved a new lemma too. In the following, we will use the new lemma to give out the main result. In chapter4, we will study the Hbpf, homoclinic and heteroclinic bifurcation of the system we are going to study. Using the Hbpf, homoclinic and heteroclnic bifurcation theory, we study the following polynomial system:$$\left\{\begin{array}{lll}\dot{x}=y,\\dot{y}=-（x^5-\frac{5}{2}x^3+x）-\epsilon（a₀+a₁x^2+a₂x^4+a₃x^6+a₄x^8）y,\end{array}\right.\eqno（1）$$where$0<\epsilon\lll$, $\delta=（a₀,a₁,a₂,a₃,a₄）\in D\subset R^5$, with$D$is bounded.System$（1.3₀）$in$[25]$under9purturbation is just system$（1）$. To thesystem$（1.3₀）$in$[25]$under5degree purturbation$（1.3_\epsilon）$:$$\left\{\begin{array}{lll}\dot{x}=y,\\...