Invariant curves and integrability of planar \(\mathcal C^r\) differential systems
type: Item
instance of: scholarly article
title: Invariant curves and integrability of planar \(\mathcal C^r\) differential systems
zbMATH Open document ID: 1401.34003
DOI: 10.1216/RMJ-2018-48-5-1537
author: Antoni Ferragut
author: Jaume Llibre
published in: Rocky Mountain Journal of Mathematics
publication date: 2018-10-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1539936035
review text: This paper considers \({\mathcal C}^r\) real planar differential systems of the form \[ \dot{x} = P(x,y),\;\;\dot{y}=Q(x,y), \] where \((x,y) \in D \subseteq \mathbb{R}^2\), \(D\) is the domain of definition of the system and \(P,Q\in {\mathcal C}^r\) where \(r\) is a positive integer, or \(r = \infty\), or \(r = \omega\) (meaning that the system is analytic). This paper improves the known expression of the \({\mathcal C}^r\) differential systems in the plane having a given \({\mathcal C}^{r+1}\) invariant curve, or a given \({\mathcal C}^{r+1}\) first integral. Application of the expressions to polynomial systems also improve known results on polynomial differential systems having either an invariant algebraic curve, or a first integral. The main theorem shows that if a polynomial differential system has an invariant algebraic curve \(g = g(x,y) = 0\), and \(g_x \not\equiv 0\), then the system has the normal form \[ \dot{x} = A g - B g_y, \;\;\;\dot{y} = B g_x, \] where \(A\) and \(B\) are suitable rational functions.
reviewed by: Jinzhi Lei
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 6958791
zbMATH Keywords: planar differential system
zbMATH Keywords: \({\mathcal C}^r\) function
zbMATH Keywords: invariant curve
zbMATH Keywords: Darboux theory
zbMATH Keywords: integrability
MaRDI profile type: Publication
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OpenAlex ID: W2897086140
description: scientific article; zbMATH DE number 6958791