Limit Cycle Problems For Some Kinds Of Planar Polynomial Systems

In this thesis, we investigate the limit cycle problem into two main topics of some planar autonomous quadratic systems and cubic systems. The major part of this thesis is devoted to the study of type â…¢ quadratic systems with m = 0 in Ye's classifications.The thesis begins with chapter 1 as an introduction, we provide a review of some concepts in qualitative theory of polynomial differential system as well as the theory of limit cycles, and some Lemmas to be use later are concluded.Chapter 2 deals with the study of two types of non Lienard cubic systems. We proof the uniqueness of limit cycle in most cases. For the second system, we find an example to show that the center conditions of quadratic systems given in [9], [17]. [49] are not complete.The main contribution of chapter 3 is to study a general cubic Lienard system with one antisaddle and two saddles. We discuss the nonexistence, existence and uniqueness of limit cycles for the system.In chapter 4 we study systematically the limit cycle problem for type â…¢ quadratic system of Ye's classifications with m = 0. First we prove that the limit cycle is center distributed only around one focus. Then without loss of generality, we can discuss the limit cycles around critical point O. The system to be considered can be takenin the formIt is easy to see that the system has no limit cycle as δ = 0 and it forms a generalized rotated vector field with respect to parameter δ in the part 1 + ax - y > 0, for which the limit cycle my appear around O. Since there is at most one limit cycle may created from O by the Hopf bifurcation, the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through the saddle S₁ on 1 + ax - y = 0 and in the right half plane, which has the same stability with the limit cycle created by the Hopf bifurcation. then we prove the limit cycle is at most one in such cases, which are decided by section conditions on the parameters l, n, a. If it is a homoclinic cycle passingthrough the saddle A"(0. -) (ri =t 0). which has the different stability with the limit cycle created by Hopf bifurcation, then it appears the two limit cycles case (also decided by some conditions on /, v and a). For the case that the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity, then it appears also two limit cycles case. For the two later cases, our discussion shows that the number of limit cycles will appear the process changing from one to two depending on the different values of /. n and a for such cases. The different topological structures related to the limit cycles around O are also given.