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

 Data: 2014 Resum: 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. Nota: Número d'acord de subvenció MICINN/MTM 2008–03437 Nota: Número d'acord de subvenció AGAUR/2009/SGR-410 Nota: Agraïments: The second author is supported by AGAUR grant PIV-DGR-2010 and by FCT through CAMGDS, Lisbon. Drets: Tots els drets reservats. Llengua: Anglès. Document: article ; recerca ; submittedVersion Matèria: Liouvillian integrability ; Invariant ellipse ; Quadratic planar polynomial vector fields Publicat a: Acta Mathematica Sinica. English Series, Vol. 30 Núm. 3 (2014) , p. 453-466, ISSN 1439-7617

DOI: 10.1007/s10114-014-2484-1

 Preprint 15 p, 739.0 KB

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