The Properties Of A Class Of Quasi-homogeneous Vector Fields

Planar polynomial differential system plays an extremely important role in the discipline of population ecology, life science and biological chemistry.In terms of the-ory, or from the method has fruitful results, for the purposes of some of the more special systems, relatively easy to study, generally get the results accomplished, and some literature as [54-57] studied from different angles for different types of homo-geneous and quasi-homogeneous differential system, accordingly we got some good conclusions. But for higher order differential systems, because of the limitation of the method and the complexity of the operation, so far there is no complete results.In this paper, we study the global topological equivalence classes of a class of homoge-neous planar vector fields, whose global properties are determined by their geometric properties of infinity.We first study such quasi homogeneous vector field views cut plane vector fields and vector fields induced by nature, drawing on the literature [1] in the stream on in our study of times quasi-homogeneous vector field plane the flow is topologically equivalent to the conclusion, as well as such times quasi-homogeneous vector field on the plane of infinity stream in the unit circle tangent vector field on the flow is topologically equivalent to the conclusion, and reference literature [1] ap-proach, using a central projection for this type of thinking times quasi-homogeneous vector field of the second plane nature of a comprehensive in-depth study to prove that this kind of plane geometric properties of quasi-homogeneous vector field de-pends only in its induced vector field.Based on the study of the basic properties of tangent vector field induction vector field, according to the nature of the induced vec-tor field it is proved that the vector field has 10 different topologies of the same sector area, given the same region for each sector the detailed classification, and draw the corresponding diagram, and then depending on the situation specific analysis, classi-fication discussed the global vector field topology,proving that the vector field a total of 49 different topological structures in the phase diagram, which when is even, there are 17 kinds of different global topological equivalence class; when is odd,32 kinds of different global topological equivalence class, and draw the corresponding global phase.