Nonintegrability of a Halphen system
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Departamento de Matemática)

Date:	2014
Abstract:	In this paper we study the Halphen system given by x˙1 = x2x3 - x1(x2 + x3) - α21(x1 - x2)(x3 - x1), x˙ 2 = x3x1 - x2(x3 + x1) - α22(x2 - x3)(x1 - x2), x˙ 3 = x1x2 - x3(x1 + x2) - α23(x3 - x1)(x2 - x3), where x1, x2, x3 are real variables and α1, α2, α3 are real constants. We show that when (α1, α2, α3) ∈ R3 \ {(0, 0, 0)} then it does not admit any first integral that can be described by formal power series. As a corollary we also obtain the nonexistence of analytic first integrals. Furthermore, we also prove that it does not admit any first integral of Darboux type.
Note:	Agraïments: Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundaçâo para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003.
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Analytic first integrals ; Halphen system
Published in:	Theoretical and Mathematical Physics, Vol. 181 Núm. 2 (2014) , p. 1383-1395, ISSN 1573-9333

DOI: 10.1007/s11232-014-0219-1

Postprint 15 p, 712.0 KB

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