Global centers of a class of cubic polynomial differential systems

Abstract

A difficult classical problem in the qualitative theory of differential systems in the plane ${\mathbb{R}}^2$ is the center-focus problem, i.e. to distinguish between a focus and a center. Another difficult problem is to distinguish inside a family of centers the ones which are global. A global center is a center $p$ such that ${\mathbb{R}}^2\setminus \{p\}$ is filled with periodic orbits. In this paper we classify the global centers of the family of real polynomial differential systems of degree $3$ that in complex notation write $$i\dot{w}=w-A_3{\overline{w}}^2-A_4{w}^3-A_5{w}^2\overline{w}-A_{6}w{\overline{w}}^2,$$ where $w=x+iy$ and $A_k\in \mathbb{C}$ for $k=3,4,5,6$.