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

Jaume Llibre, Yilei Tang, Jiang Yu, Pengyu Zhou

TL;DR

The paper addresses the global dynamics of planar quadratic and cubic quasi-homogeneous but non-homogeneous polynomial differential systems by showing that such systems can be reduced to a finite set of homogeneous cases via weight-based transformations, then analyzed with blow-up, normal-sector, and Poincaré-compactification techniques. It derives canonical homogeneous forms, classifies the associated dynamics at infinity, and provides a complete taxonomy of global phase portraits for both degrees, including explicit parameter conditions that realize each portrait. The main contributions are the reduction to homogeneous families $H_2,H_1,H_0$ and the explicit global-portrait classification (for both quadratic and cubic cases) along with the corresponding invariant structures and separatrix configurations. This work extends classical homogeneous-system results to the broader quasi-homogeneous setting, offering precise, parameter-dependent portraits that can inform applications in control, biology, and economics where multi-scale nonlinear dynamics arise.

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, Poincaré 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.

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

TL;DR

The paper addresses the global dynamics of planar quadratic and cubic quasi-homogeneous but non-homogeneous polynomial differential systems by showing that such systems can be reduced to a finite set of homogeneous cases via weight-based transformations, then analyzed with blow-up, normal-sector, and Poincaré-compactification techniques. It derives canonical homogeneous forms, classifies the associated dynamics at infinity, and provides a complete taxonomy of global phase portraits for both degrees, including explicit parameter conditions that realize each portrait. The main contributions are the reduction to homogeneous families and the explicit global-portrait classification (for both quadratic and cubic cases) along with the corresponding invariant structures and separatrix configurations. This work extends classical homogeneous-system results to the broader quasi-homogeneous setting, offering precise, parameter-dependent portraits that can inform applications in control, biology, and economics where multi-scale nonlinear dynamics arise.

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, Poincaré 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.

Paper Structure

This paper contains 5 sections, 13 theorems, 25 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-homogeneous polynomial systems of degree 2 and 3
  4. The corresponding homogeneous polynomial systems
  5. Global dynamics and phase portrait analysis

Key Result

Lemma 1

For the singularity $E=(0, u_0)$ of system eqn-7, the following statements hold: (a) For $G'(1, u_0) \neq 0$, the singularity $E$ is - either a saddle if $P_{n}( 1, u_{0}) G^{\prime }( 1, u_{0}) < 0$, - or a node if $P_{n}( 1, u_{0}) G^{\prime }( 1, u_{0}) > 0.$ (b) For $G'(1, u_0)=0$, if $u_0$ is a

Figures (7)

  • Figure 1: Three classes of normal sectors
  • Figure 2: Portraits of $\tilde{(3d)}$ when $\widehat{G}_{2}(u)$ has two different zeros
  • Figure 3: Portraits of $\tilde{(3d)}$ when $\widehat{G}_{2}(u)$ has one zero of multiplicity 2
  • Figure 4: Portraits of $(3d)$ when the infinity has 2 singularities
  • Figure 5: Portraits of $(3d)$ when the infinity fulfills singularities
  • ...and 2 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • proof
  • Theorem 9
  • ...and 7 more