Global dynamics of planar quasi-homogeneous differential systems

Abstract

In this paper we provide a new method to study global dynamics of planar quasi-homogeneous differential systems. We first prove that all planar quasi-homogeneous polynomial differential systems can be translated into homogeneous differential systems and show that all quintic quasi-homogeneous but non-homogeneous systems can be reduced to four homogeneous ones. Then we present some properties of homogeneous systems, which can be used to discuss the dynamics of quasi-homogeneous systems. Finally we characterize the global topological phase portraits of quintic quasi-homogeneous but non-homogeneous differential systems.