Hopf Bifurcation For Two Types Of Liénard Systems And Bifurcation Near A Cuspidal Loop In A Near-Hamiltonian System

The second part of Hilbert's 16th problem is related with the maximal number and relative positions of limit cycles of polynomial systems of degree n. Many many works have been done on the study of the above problem for many years, especially for quadratic and cubic systems. However, up to now the problem has been not solved com-pletely yet even for the case of n= 2. Limit cycles are generated through bifurcation, such as Hopf bifurcation, homoclinic bifurcation, heteroclinic bifurcation, Poincare bi-furcation and so on. In recent years, the study of limit cycles of nonsmooth dynamical systems has also being developed. Some general fundamental results of bifurcation for nonsmooth system were got.This paper consists of five chapters. The particular contents of each chapter are as follows.As an introduction, in the first chapter the background of our research and main topics, which we will study in the following chapters, are introduced. A description of our methods and results derived in this thesis can be found in this chapter.Chapter 2 is related with preliminary lemmas. Our main purpose is to provide a detailed proof of several main lemmas which play an important role in proving main results.In Chapter 3, we study the Hopf bifurcation of two types of smooth Lienard poly-nomial systems. First, we use a new method to prove that for the Lienard polynomial systemwhere qn(x) is a polynomial in the variable x of degree n and qn(0)= 0, the Hopf cyclicity near a center is Second, by applying the new method to another Lienard polynomial systemwhere pm(x) is a polynomial in the variable x of degree m and pm(0)≠0, we prove that for Eq.(2) the upper bound of the maximal number of local limit cycles is Further, we obtain that the Hopf cyclicity of Eq.(2) near the center is 2n - 2 for m=n=1,2,3,4.In Chapter 4, we study the Hopf cyclicity of nonsmooth polynomial system. By applying the methods in the second and third chapter, we consider the nonsmooth polynomial system where and prove that the Hopf cyclicity at the origin is for m≥n or for n>m.In the last chapter, we investigate one kind of near-Hamiltonian system which has a cuspidal loop and where P3(x, y) and Q3(x,y) are polynomials in the variable x and y of degree 3. And we prove that the number of limit cycles appearing in a neighborhood of the cuspidal loop is 5, by using some known bifurcation theorems to study the first Melnikov function.The main method we use in this paper is that, during the study of Eq.(1), by mak-ing variable transformation we prove the linear independence to get the condition which is necessary to complete the proof of main results. It is different from the method of complex analysis used in Petrov [11]. And the method also can be implied to polynomial system (2) and nonsmooth Lienard system (3).