Home > Articles > Published articles > Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse |
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 |
Postprint 15 p, 739.0 KB |