New families of global cubic centers

Abstract

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