New families of global cubic centers

An equilibrium point p of a differential system in the plane \(\mathbb {R}^2\) ${\mathbb{R}}^2$ is a center if there exists a neighbourhood U of p such that \(U\setminus \{p\}\) $U\backslash \{p\}$ is filled with periodic orbits. A difficult classical problem in the qualitative theory of differential systems in the plane \(\mathbb {R}^2\) ${\mathbb{R}}^2$ is the problem of distinguishing between a focus and a center. A global center is a center p such that \(\mathbb {R}^2 \setminus \{p\}\) ${\mathbb{R}}^2\backslash \{p\}$ is filled with periodic orbits. Another difficult problem in the qualitative theory of differential systems in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ is to distinguish inside a family of centers the ones which are global. Lloyd, Pearson and Romanovsky characterized when the origin of coordinates is a center for the family of cubic polynomial differential systems \(\begin{array}{*{20}l} {\dot{x} = y - Cx^{2} + \left( {B + 2D} \right)xy + Cy^{2} + Px^{3} + Gx^{2} y - \left( {H + 3P} \right)xy^{2} + Ky^{3} ,} \hfill \\ {\dot{y} = - x + Dx^{2} + \left( {E + 2C} \right)xy - Dy^{2} - Kx^{3} - \left( {H + 3P} \right)x^{2} y - Gxy^{2} + Py^{3} .} \hfill \\ \end{array}\) $\begin{array}{c}\dot{x}=y-C{x}^2+\left(B+2D\right)xy+C{y}^2+P{x}^3+G{x}^{2}y-\left(H+3P\right)x{y}^2+K{y}^3,\hfill \\ \dot{y}=-x+D{x}^2+\left(E+2C\right)xy-D{y}^2-K{x}^3-\left(H+3P\right)x^{2}y-Gx{y}^2+P{y}^3.\hfill \end{array}$ . Here we characterize when the origin of this family of differential system is a global center.