The Properties Of A Class Of The Cubic Quasi-homogeneous Vector Fields

No matter the, linear or no-linear multinomial ordinary differential dynamical systems do they all have widespread application in biochemistry, life science ecology areas etc. Many scholars all over the world have researched on.it,and they also have achieved some wonderful results that relate to global topological structure of some special classes of dynamical systems, such as square subsystems that have no common factor on the right and possess star type point, do not have limited cycle,only contain the saddle points, points and half saddle nodes during their limited singular points and possess17kinds of the overall situation phase diagram. Author had researched the global topological structure of a class of cubic differential system possessing star type points, showed that it existed26kinds of different global topological classifications in literature [6]. In this paper, we discuss the topological structure in the limited domain, the properties of a class of cubic multinomial differential system and the properties of the equatorial closed orbit. The main method we used,developed by the author in the [1] are suitable to research the topological relationship between the flow of the quadric vectorfield and the induced vectorfield. We just discuss the topological of our dynamical system in the limited domain, since the flow of the quadric vectorfield and the induced vectorfield are not topological equivalence in the infinite distance,and obtain15kinds of different topological phase portraits.