Investigation For A Class Of Vector Fields In R~3

In this paper, a bridge between quasi-homogeneous vector fields and homogeneous vector fields in R3 has been found:the induced tangent vector fields of two-dimensional manifold S 2 are the same, and invariant cones with the vertex at the origin of the homoge-neous vector fields must be ones of quasi-homogeneous vector fields. By the relationship, we change the investigation of complex quasi-homogeneous vector fields to the investi-gation of homogeneous vector fields. At present, for the homogeneous vector fields in R3, the most results are about twice homogeneous vector fields. At first, on base of [1], bridges between twice homogeneous vector fields and induced tangent vector fields, and between tangent vector fields and induced vector fields in R2 are found, so we change the investigation of complex quasi-homogeneous vector fields to the investigation of vector fields in R2. Then we investigate another kind of the simpler quasi-homogeneous vector fields. at last, we proved the existence of closed orbits for two kinds of the simpler quasi-homogeneous vector fields in R3, when induced tangent vector fields exist closed orbits, and got many results about limit circles and homoclinic(heteroclinic)orbit.