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

Joji Benny, Supriyo Jana, Soumen Sarkar

• In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere S2 = {(x, y,z) ∈ R3 | x2 + y2 +z2 = 1}. We start by classifying all degree three polynomial vector fields on S2 and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on S2 and also study the maximum number of various types of invariant great circles for homogeneous cubic vector fields on S2. We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on S2.