On a conjecture on the integrability of Liénard systems
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Murza, Adrian C. (Insitutul de Matematică Simion Stoilow al Academiei Române)
Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)

Additional title:	We prove a conjecture on the integrability of Liénard systems
Date:	2020
Abstract:	We consider the Liénard differential systems ̇x=y+F(x), ̇y=x (1), in C2 where F(x) is an analytic function satisfying F(0) = 0 and F'(0) ≠ 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F(x) is linear, i. e. F(x) = ax. Here we prove this conjecture, and show that when F(x) is linear and system (1) has an analytic first integral, this is a polynomial.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Liénard system ; First integral ; Strong saddle
Published in:	Rendiconti del Circolo Matematico di Palermo, Vol. 69, Issue 1 (April 2020) , p. 209-216, ISSN 1973-4409

DOI: 10.1007/s12215-018-00398-6

Postprint 10 p, 300.0 KB

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