Topological Classification For A Class Of Three-degree Quasi-homogeneous Cubic Differential Systems In R~2

Some global geometric properties of planar polynomial differential systems have obtained some very good results, for example Global geometric properties of two-degree and there-degree cubic homogeneous dif-ferential systems has been perfect, but there is no study of the nature of the differ-ential systemsIn this paper, we study the topological Classification for a class of quasi-homogeneous cubic differential systems with x=Q(x)+xf(x) in R2, and x=(x1,x2), Q(x)=(Q1(x),Q1(x)), Qi(x1,x2)=a20x12+a11x1x2+a02x22, Q2(x1,x2)=b20x12+b11x1x2+b02x22, f(x1,x2)=ax12+bx1x2+cx22are all two-degree homogeneous polynomial. Some properties of cubic differential systems are studied firstly in R2, and then the properties of the tangent vector field and its induced vector field are shown, too, Use the method of topological equivalent relationship between the flow of the vector field and the differential systems of this paper by the literature[1] and [2]. Because the differential systems of this paper and its geometric properties at in-finity are not topological equivalent, so we only investigate the topology of the origin localized within the system trajectories, and give the classification of its topological structure, and then we prove that it exactly has21kinds of different topological phase diagrams.