Global Topological Classification And Coefficient Conditions Of A Kind Of Plane Homogeneous Fifth System

In this dissertation, we study the global topological classification and coefficient conditions of the plane homogeneous fifth polynomial differential systemThe main techniques used in this thesis includes the methods of the global structure and coefficient conditions of the plane homogeneous quadratic and cubic system mentioned in the paper [1] of professor Ye yanqian, and the paper [2] of professor Li xue min, also includes the idea to high-order critical point of professor Zhang zhifen, Lu yulin and Han yuliang etc. Due to the degree of polynomial in the right of equal-sign crease, when we discuss the global structure, the more special directions, the more difficulty in drawing phase portraits of this system. Finally, we also discuss the globally asymptotically stability of this system by consulting the paper [3]. The discussing process follows the steps below:First, suppose the system (1) has only one finite singular point (0,0).then we canassume b50 = 0, which special direction is determined by equation G(0) -0,introduce Poincare transformation to discuss infinite singular points, according to the coefficient conditions, list all possible infinite singular points and special directions , judging their type, drawing out all kinds of phase portraits.Second, give some examples, make them have one to six special direction, through solve these examples, demonstrate the conclusion of these theories in chapter 2 are c-orrectly.Last, according to the trend of the trajectory we can determine the boundedness of the system and globally asymptotically stability of the zero solution.