Homogeneous polynomial vector fields of degree 2 on the 2―dimensional sphere

Abstract

Let X be a homogeneous polynomial vector field of degree 2 on 𝕊² having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of 𝕊², and at most there are two invariant circles. We characterize the global phase portrait of these vector ¯elds. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.