Geometric Properties Of Half Quasi-homogeneous Vector Field

Most research in the qualitative theory of ordinary differential equation stay in the plane field,since we have Poincare-Bendixsion theory.There are few methods can be used in the systems in three dimension.We can just deal with some special equations.This paper give some properties about a special one called Quasi-homogeneous vector field.Coleman,Camacho,Sharipov and Liang Zhaojun have done well in one type of system names Homogeneous vector field. By using the famous Poincare transformation, we can change the original system into two related systems.One tangent vector field induced in the ball.Then we can use theories in the plane.Since we can get convenient from the transformation, the later work was to find more systems which can be changed by the same method. It was my teacher Professor Zhang Xing’an first find the socalled Quasi-homogeneous vector field which can also be researched well in the same way, and this type is different from Homogeneous vector field.Professor Zhao Yulin also found another type of system which can be transformed in the same way,and also named Quasi-homogeneous vector field.To distinguish from each other,we called the system found by Professor Zhang in Chinese way.In this paper, we use the same way in which Professor Zhang find the Quasi-homogeneous vector to explore the vector field found by Professor Zhao.And we find a new type system and the invariant surface, periodic solution in three dimensional space. This is the major part of this paper.Since there are some differences between the two ways made by Professor Zhao and by Professor Zhang,writer found a special example,which fulfill the two definitions.Then we can research it in two ways,and get more properties compared to one method.