Invariant Circles and Phase Portraits of Cubic Vector Fields on the Sphere

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