Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Departamento de Matemática)

Date:	2014
Abstract:	We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse. More precisely, a quadratic system having an invariant ellipse can be written into the form ˙x = x2+y2−1+y(ax+by+c), ˙y = −x(ax+by+c), and the ellipse becomes x2+y2 = 1. We prove that (i) this quadratic system is analytic integrable if and only if a = 0; (ii) if x2 + y2 = 1 is a periodic orbit, then this quadratic system is Liouvillian integrable if and only if x2 + y2 = 1 is not a limit cycle; and (iii) if x2 + y2 = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0.
Grants:	Ministerio de Ciencia e Innovación MTM 2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410
Note:	Agraïments: The second author is supported by AGAUR grant PIV-DGR-2010 and by FCT through CAMGDS, Lisbon.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Liouvillian integrability ; Invariant ellipse ; Quadratic planar polynomial vector fields
Published in:	Acta Mathematica Sinica. English Series, Vol. 30 Núm. 3 (2014) , p. 453-466, ISSN 1439-7617

DOI: 10.1007/s10114-014-2484-1

Postprint 15 p, 739.0 KB

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