Phase portraits of quadratic systems with finite multiplicity three and a degenerate critical point at infinity
type: Item
instance of: scholarly article
title: Phase portraits of quadratic systems with finite multiplicity three and a degenerate critical point at infinity
zbMATH Open document ID: 0896.34022
author: J. W. Reyn
author: Xianhua Huang
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-10-06T00:00:00Z
full work available at URL: CONT27-3.html
review text: The following two theorems are proved. Theorem 1. Any quadratic system with three finite critical points (complex or coinciding) and an \(M^i_{1,3}\) type of infinite critical point can be transformed by a linear transformation into \[ \dot x=\lambda x+\mu y+ xy+ y^2,\quad \dot q=x+ y^2. \] There are 22 topologically different phase portraits; a limit cycle, if it exists, must be unique and hyperbolic. Theorem 2. Any quadratic system with three finite critical points and an \(M^i_{1,2}\) type of infinite critical point can be transformed into \[ \dot x= \lambda x+\mu y+ \gamma xy+\delta(x+ y^2),\quad \dot y= x+y^2,\;\lambda\in\{0,1\},\;\gamma\not\in\{0, 1\}. \] There are 119 topologically different phase portraits; a limit cycle, if it exists, must be unique and hyperbolic.
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 1143476
zbMATH Keywords: quadratic systems
zbMATH Keywords: critical points
zbMATH Keywords: phase-portrait
zbMATH Keywords: limit cycles
zbMATH Keywords: degenerate critical point at infinity
reviewed by: Yanqian Ye
MaRDI profile type: Publication
OpenAlex ID: W1996572656
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DOI: 10.1216/RMJM/1181071903
description: scientific article; zbMATH DE number 1143476