Global dynamics of planar quasi-homogeneous differential systems

TL;DR

This paper introduces a new method for analyzing the global dynamics of planar quasi-homogeneous differential systems by translating them into homogeneous systems and characterizing their phase portraits.

Contribution

It provides a novel approach to reduce quasi-homogeneous systems to homogeneous ones and characterizes their global phase portraits, especially for quintic cases.

Findings

All planar quasi-homogeneous polynomial systems can be transformed into homogeneous systems.

Quintic quasi-homogeneous non-homogeneous systems reduce to four homogeneous systems.

Characterization of global phase portraits for certain quintic 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.