In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere $\mathbb{ S}^2 = \{ (x, y, z) \in \mathbb{ R}^3 ~|~ x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector fields on $\mathbb{ S}^2$ and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on $\mathbb{ S}^2$ and also study the maximum number of various types of invariant circles for homogeneous cubic vector fields on $\mathbb{ S}^2$. We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on $\mathbb{ S}^2$.
Categories: math.DS math.CA