Polynomial first integrals of quadratic vector fields
type: Item
instance of: scholarly article
title: Polynomial first integrals of quadratic vector fields
zbMATH Open document ID: 1117.34035
author: Javier Chavarriga
author: Belén García
author: Jaume Llibre
author: J. A. Rodríguez
author: Jesús S. Pérez del Río
published in: Journal of Differential Equations
publication date: 2007-01-09T00:00:00Z
review text: The goal of this paper is to characterize those quadratic differential equations in the plane having a polynomial first integral, and to provide an explicit expression of these systems as well as of their polynomial first integrals. It is well known that if the system writes as \[ \dot x=P(x,y), \quad \dot y=Q(x,y) \] it is not restrictive to assume that \(P(x,y)\in\{1+xy, xy, y+x^2, y, 1+x^2, -1+x^2, x^2, x, 1, 0\}\) and \(Q(x,y)=a+bx+cy+dx^2+exy+fy^2.\) The authors consider only the cases where the system has finitely many critical points and solve completely the problem. For instance, their result when \(P(x,y)=1+x^2\) says that in this case the system has a polynomial first integral \(H(x,y)\) if and only if, either \(f=c=b=d=0,\) \(e=-2k\) with \(k\) a positive integer and \(a=1,\) and then \[ H(x,y)=y(1+x^2)^k-\sum_{j=0}^{k-1}{{k-1}\choose{j}}{{x^{2j+1}}\over{2j+1}}, \] or \(f=a=c=b=d=0\) and \(e=-p/q\) being a negative rational number, and then \(H(x,y)=(1+x^2)^py^{2q}.\) The more complicated cases in their study are the more generic ones, \textit{i.e.} \(P(x,y)=1+xy\) and \(P(x,y)=xy.\)
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 5083413
zbMATH Keywords: quadratic differential equation
zbMATH Keywords: polynomial first integral
MaRDI profile type: Publication
full work available at URL: j.jde.2006.07.022
OpenAlex ID: W1992378476
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DOI: 10.1016/J.JDE.2006.07.022
description: scientific article; zbMATH DE number 5083413