Invariant algebraic curves of large degree for quadratic system
type: Item
instance of: scholarly article
title: Invariant algebraic curves of large degree for quadratic system
zbMATH Open document ID: 1076.34030
author: Colin Christopher
author: Jaume Llibre
author: Grzegorz Świrszcz
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-07T00:00:00Z
review text: This paper presents several interesting examples of quadratic systems (QS) having invariant algebraic curves of large degree. These examples are: (i) QS having algebraic limit cycles of degree 5 and 6; (ii) QS having algebraic saddle loops of degree 3 and 5 surrounding a strong focus; (iii) the QS, \(\dot x=1+x^2+xy,\,\) \(\dot y=57/2-81/2x^2+3y^2,\) which is not Liouvillian integrable and has an invariant algebraic curve of degree 12. It is worth to mention that until this paper appeared, the highest degree known for algebraic limit cycles of QS was 4. The new examples having limit cycles of degrees 5 and 6 have been constructed following the next clever idea: By applying to a QS of the form \[ dx/dt=\alpha x+\beta y+2ex^2+bxy+cy^2,\quad dy/dt=\gamma x+\delta y+ exy+fx^2, \] the transformation \(X=x/y^2, Y=1/y\) and the change of time \(dt/dT=Y,\) the new system still remains in the class of QS. Then, starting from the known QS having algebraic limit cycles of degree 4, new systems having also algebraic limit cycles of degrees 5 and 6 are constructed. It is unknown whether there are examples of QS having algebraic limit cycles of degree higher than~6.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 2153754
zbMATH Keywords: quadratic system
zbMATH Keywords: algebraic limit cycle
zbMATH Keywords: algebraic loop
zbMATH Keywords: Liouvillian integrability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.08.042
OpenAlex ID: W2051773597
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DOI: 10.1016/J.JMAA.2004.08.042
description: scientific article; zbMATH DE number 2153754