Two Types Of Polynomial System

In this paper,Based on the qualitative theory and the bifurcation theory,the qualitative, the Hopf bifurcation,as well as the global structure questions of two types of planar polynomial systems are studied.The full text of the content is divided into four chapters.The first chapter is the preface,the history and development of planar polynomial differential system with limit cycle and bifurcation are introduced.Some basic concepts and lemmas of the stability theory and bifurcation theory in this paper are given,and the main job are easy introduced.The second chapter,limit cycle distribution and the Hopf bifurcation of a class of quadratic differential system are studied.Some conditions about the nonexistence of limit cycles and the limit cycle distribution are fixed by the Dulac discriminance.What is decided that the origin is the first-order focus by the form of series method.The existence,uniqueness and stability of limit cycles are studied by the Hopf bifurcation.The third chapter spread the quadratic differential system in the second chapter.By analyzed qualitative of the singular points,the distribution of limit cycles are fixed.And the existence,uniqueness and stability of limit cycles are studied by the Hopf bifurcation. Conclusions have been popularized.The fourth chapter,all of the singular points of a class of E₃¹ system when a₂=b₄= 0,and a₁≠0,a₃≠0 are discussed.Sufficient condition for a series of the Hopf bifurcation are gave.And all the possible global structures of the system are obtained by using some methods to analyze the infinite singular points when the origin is center.