Bifurcation Of Limit Cycles Of Polynomial Systems

In this dissertation, by using the method of qualitative analysis and bifurcaton theory, we investigate the number and distribution of limit cycles for some planar ploynomial systems under perturbation with the help of some mathematical tools such as Maple 10.0. The study develops the method of stability-changing to investigate the bifurcation of limit cycles. We also obtain the expansion of the first Melnikov function for a system exhibiting a cuspidal loop. We study cuspidal loop bifurcation, Homoclinic or double Homoclinic bifurcation, Hopf bifurcation and obtain the necessary conditions to generate limit cycles by using the coefficients of the expansion of the first Melnikov function. By computing the focus values, we give out conditions that a system has centers and also investigate the highest order of a focus point. This paper is divided into seven parts.In Chapter 1, we study the expansion of the first Melnikov function for the general Hamiltonian sytem exhibiting a cuspidal loop. Then we investigate the necessary and sufficient conditons for a kind of Hamiltonian system which is symmetric with respect to the x-axis exhibiting a cuspidal loop. And as an application, we study the bifurcation of a cuspidal loop.Chapters 2 and 3 concern the limit cycles bifurcated from the focus and the homoclinic loop for a kind of near-Hamiltonian systems by taking advange of the coefficients appeared in the first order Melnikov function. We obtain a new sufficient condition to find limit cycles in global bifurcations.Chapter 4 investigates the number and distributions of limit cycles for a class of Kukles system, which also can be regarded as a class of reduced Kukles system under cubic perturbation. Using the techniques of bifurcation theory and qualitative analysis, we have obtained three different distributions of five limit cycles for the considered system. In the first two distributions, the five limit cycles are all non-small amplitude, which is very different from the previous work.Chapter 5 concerns the number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact of there exist 9-11 limit cycles is verified. The different distributions of limit cycles are given by using the methods of bifurcation theory and qualitative analysis, among which two distributions of eleven limit cycles are new.Chapter 6 is related to the study of the number of limit cycles for a quintic near-Hamiltonian system. It is verified that the system can have 20,22,24 limit cycles with different distributions of limit cycles for each case. The limit cycles are obtained by using the methods of stability-changing. However, for the same unperturbed system, after a different perturbation, other authors can only get 23 limit cycles.As we know, Bautin proved that the highest order for a focus of a quadratic system is 3. In chapter 7, We give a new and simple proof to this well-known result by using an elementary method. We also use the elementary method to give a necessary and sufficient condition for a quadratic system to have a center.The main results of this dissertation are illustrated by some examples.