Centers of cubic polynomial differential systems

Abstract

An equilibrium point of a differential system in the plane is a center if there exists a neighborhood of such that is filled with periodic orbits. A difficult classical problem in the qualitative theory of differential systems in the plane is the problem of distinguishing between a focus and a center. In this paper we characterize when the origin of coordinates is a center of the following cubic polynomial differential systems where is an arbitrary nonzero monomial of degree 3. Moreover we provide all topologically different phase portraits when .