Existence of limit cycles for real quadratic differential systems with an invariant cubic
type: Item
instance of: scholarly article
title: Existence of limit cycles for real quadratic differential systems with an invariant cubic
zbMATH Open document ID: 1116.34025
author: Javier Chavarriga
author: Isaac A. García
published in: Pacific Journal of Mathematics
publication date: 2007-02-26T00:00:00Z
review text: A quadratic system is a system of ordinary differential equations of the form \[ \dot x = P(x,y),\quad \dot y = Q(x,y) (QS), \] where \(P\) and \(Q\) are relatively prime polynomials for which \(\text{max}\{\deg P, \deg Q \} = 2\). An invariant cubic is an algebraic curve \(f(x,y) = 0\), where \(f\) is a cubic polynomial, which is composed of orbits in the phase portrait of system (QS). Any quadratic system that has an irreducible cubic invariant curve can, by an affine change of coordinates and a time rescaling, be placed in one of twelve canonical forms, for each of which the form of \(f\) is known. In this paper the authors show that in exactly two of these classes of canonical forms is a limit cycle in the phase portrait possible, and give an example of a canonical system with a limit cycle in each of these two cases. (It is impossible that the cycle in question actually lie in the cubic curve. The authors give a new and the simplest known proof of this well-known fact.)
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 5129577
zbMATH Keywords: limit cycle
zbMATH Keywords: invariant algebraic curve
zbMATH Keywords: quadratic system
MaRDI profile type: Publication
full work available at URL: pjm.2006.223.201
OpenAlex ID: W2131516066
DOI: 10.2140/PJM.2006.223.201
description: scientific article; zbMATH DE number 5129577