Qualitative Analysis For A Class Of Polynomial Differential System

In this dissertation,we mainly discuss the qualitative analysis and the distributions of limit cycles of a special K?model of polynomial differential system. First,we discuss the qualitative analysis of the system:Where PS (x,y) and Q4(x, y) have no common factor. We obtain some corresponding phase portraits of the system.The main techniques used in this thesis include the idea of algebraic classification of Llibre,and the ideas of high-order critical point of professor Zhang Zhifen ?Li Xuemin , Hu Qinxun etc. By means of the theory of algebraic invariant,we obtain canonical forms of the fourth-order binary forms on the real domain.Then on basis of this,we obtain the algebraic classification of the systmThere're ten topological equivalent classes.So we only need study them.Second, we also discuss the distributions of limit cycles of a class of codimension-two degenerate highter power plane polynomial differential system:Where P(x, y) and Q(x, y) are polynomials of degree not less than 5.We reduce the vector field by normal form theory.Then the study is equivalent to the study of the system:which is a class of 5-degree Lienard equations with central symmetry.The main techniques used in this part include the ideas of Wang Mingshu ? Luo Dingjun?Li Xuemin , Suo Guangjian etc, which are used to study the global analysis forn = 3.We also use the result of Zhou Yurong , Han Maoan,which is the uniqueness and duality oflimit cycles surronding several singular points. As a result,we obtain the distributions of limit cycles.