Global dynamics of planar quasi-homogeneous differential systems

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.