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 $$ i\dot{w}=w-A_3\overline{w}^2-A_4w^3-A_5w^2\overline{w}-A_6w\overline{w}^2, $$ where $w=x+iy$ and $A_k\in\mathbb{C}$ for $k=3,4,5,6$.