Linear type global centers of linear systems with cubic homogeneous nonlinearities

Abstract

A center p of a differential system in $$\mathbb {R}^2$$ is global if $$\mathbb {R}^2 \setminus \{p\}$$ is filled of periodic orbits. It is known that a polynomial differential system of degree 2 has no global centers. Here we characterize the global centers of the differential systems $$\begin{aligned} {\dot{x}} = ax + b y +P_3(x,y), \quad {\dot{y}} = c x + d y + Q_3(x,y), \end{aligned}$$ with $$P_3$$ and $$Q_3$$ homogeneous polynomials of degree 3, and such that the center has purely imaginary eigenvalues, i.e. a linear type center.