A Class Of Plane Differential Dynamical Systems Qualitative Analysis And Topology

In this dissertation, we study the boundedness and unboundedness of a class of cubic svstemand obtain the coefficient condition of the bounded system.The main techniques used in this thesis includes the idea of algebraic classification of Llibre, the methods of boundedness and unboundedness to quadratic system in the plane mentioned in the papers of professor Ye yanqian,Yang xinan etc and the idea to high-order critical point of professor Zhang zhifen,Li xuemin and Lu yulin etc. The discussing process follows the steps below:First,by means of the theory of algebraic invariant,we make algebraic classification to binary quadratic and quatic homogeneous polynomial , as illustrated in Lemma2.1.1,Lemma2.1.2;Then on the basis of this,we classify the linear system and plane cubic homogeneous polynomial system, ensuring the trajectory direction unchanged,and the results are given in Theorem2.2.1,Theorem2.2.2,Theorem2.2.3.Second,according to the idea of the direct sum in the space,making use of the results obtained in the first step,we can transform the system(2.2.2) into forty different equivalence classes,given in the (2.2.2') of Theorem2.2.4, as given in the below.Meanwhile the trajectory trend remains unchanged.Last, since a degree 1-(-degree 3 system can be classified into forty classes, in order to discuss the boundedness of cubic system with isolated critical points,it is sufficient to consider each of system(2.2.2').To achieve this, turn(2.2.2') to the Poincare sphere,we can discuss the critical points in the infinity. Hence according to the trend of the trajectory we can get the sufficient and necessary conditions for the boundedness of the system(2.2.2),as can be seen in Theorems.l-Theorem3.17. To the system with more than two critical points in the infinity, we only obtained the sufficient condition for the boundedness which are Theorem3.18-Theorem3.20.