Polynomial inverse integrating factors for quadratic differential systems
type: Item
instance of: scholarly article
title: Polynomial inverse integrating factors for quadratic differential systems
zbMATH Open document ID: 1203.34063
author: B. Coll
author: Antoni Ferragut
author: Jaume Llibre
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-07-13T00:00:00Z
review text: The authors consider the real planar quadratic polynomial system \[ \dot x = P(x,y), \quad \dot y = Q(x,y), \tag{1} \] where the dot denotes the derivative with respect to the time variable, and \(P,Q\) are quadratic polynomials. A function \(R(x,y)\) is called integrating factor of system (1), if \(R\) is a solution of the equation \(\text{div}(RP,RQ) = 0\); and a function \(V(x,y)\) is called the inverse integrating factor of (1) if \(V^{-1}\) is an integrating factor of (1), i.e., \(V\) is a solution of the equation \(PV_x + QV_y = V (P_x+Q_y)\). One of the main problems in the qualitative theory of planar polynomial vector fields is to characterize the integrable ones. In this paper, the authors characterize all the quadratic polynomial systems (1) having a polynomial inverse integrating factor of degree \(k>0\) and provide explicit normal forms for such systems and for their associated first integrals. Also they prove that these families of quadratic systems have no limit cycles.
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 5736416
zbMATH Keywords: quadratic differential systems
zbMATH Keywords: polynomial inverse integrating factor
zbMATH Keywords: Darboux theory of integrability
reviewed by: Alexey O. Remizov
MaRDI profile type: Publication
full work available at URL: j.na.2010.04.004
OpenAlex ID: W1990384308
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DOI: 10.1016/J.NA.2010.04.004
description: scientific article; zbMATH DE number 5736416