Global centers of a class of cubic polynomial differential systems

A difficult classical problem in the qualitative theory of differential systems in the plane \({\mathbb {R}}^2\) ${\mathbb{R}}^2$ 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 \({\mathbb {R}}^2\setminus \{p\}\) ${\mathbb{R}}^2\backslash \{p\}$ 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 \(\begin{aligned} i{\dot{w}}=w-A_3{\overline{w}}^2-A_4w^3-A_5w^2{\overline{w}}-A_6w{\overline{w}}^2, \end{aligned}\) $$\begin{array}{c}\hfill i\dot{w}=w-A_3{\overline{w}}^2-A_4{w}^3-A_5{w}^2\overline{w}-A_{6}w{\overline{w}}^2,\end{array}$$ where \(w=x+iy\) $w=x+iy$ and \(A_k\in {\mathbb {C}}\) $A_k\in \mathbb{C}$ for \(k=3,4,5,6\) $k=3,4,5,6$ .