Global Dynamics Of Quadratic And Cubic Planar Quasi-homogeneous Differential Systems

TL;DR

This paper analyzes the global dynamics and phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous planar polynomial systems, providing a comprehensive classification using advanced mathematical techniques.

Contribution

It demonstrates that all such systems can be reduced to three homogeneous systems and characterizes their global phase portraits.

Findings

Reduction of systems to three homogeneous cases

Use of blow-up and Poincaré methods for analysis

Complete characterization of global phase portraits

Abstract

In this paper we obtain the global dynamics and phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous systems. We first prove that all planar quadratic and cubic quasi-homogeneous but non-homogeneous polynomial systems can be reduced to three homogeneous ones. Then for the homogeneous systems, we employ blow-up method, normal sector method, Poincar\'e compactification and other techniques to discuss their dynamics. Finally we characterize the global phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous polynomial systems.