In this paper, a bridge between near-homogeneous and homogeneous vector fields in R 3 is found. By the relationship between homogeneous vector fields and the induced tangent vector fields of two-dimensional manifold S 2 , we prove the existence of at least 5 isolated closed orbits for a class of n + 1 (n ≥ 2) systems in R 3 , which are located on the five invariant closed cones of the system.
Lianhua Ma, Cuihong Yang , Xinan Zhang (School of Math. and Statistics, Central China Normal University, Wuhan 430079)
In this paper, .we find a bridge connecting a class of vector fields in R3 with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and.homoclinic orbits of a class of vector fields in R3. All the possible nonwandering sets of this class of vector fields fall into three classes： （i） singularities; （ii） closed orbits; （iii） graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.
