The Closed Orbits of a Class of Cubic Vector Fields in ℝ3

Abstract

In this paper, we investigate the isolated closed orbits of two types of cubic vector fields in ℝ3 by using the idea of central projection transformation, which sets up a bridge connecting the vector field X (x) in ℝ3 with the planar vector fields. We have proved that the cubic vector field in ℝ3 can have two isolated closed orbits or one closed orbit on the invariant cone. As an application of this result, we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones, and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.