On a conjecture on the integrability of Liénard systems

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...

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Autores: Llibre, Jaume|||0000-0002-9511-5999, Murza, Adrian|||0000-0001-9521-5052, Valls, Clàudia|||0000-0001-8279-1229
Tipo de recurso: artículo
Fecha de publicación:2020
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:221375
Acceso en línea:https://ddd.uab.cat/record/221375
https://dx.doi.org/urn:doi:10.1007/s12215-018-00398-6
Access Level:acceso abierto
Palabra clave:Liénard system
First integral
Strong saddle
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spelling On a conjecture on the integrability of Liénard systemsWe prove a conjecture on the integrability of Liénard systemsLlibre, Jaume|||0000-0002-9511-5999Murza, Adrian|||0000-0001-9521-5052Valls, Clàudia|||0000-0001-8279-1229Liénard systemFirst integralStrong saddleWe 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. 22020-01-0120202020-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/221375https://dx.doi.org/urn:doi:10.1007/s12215-018-00398-6reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-77278-PMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2013-40998-PAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-568open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2213752026-06-06T12:50:31Z
dc.title.none.fl_str_mv On a conjecture on the integrability of Liénard systems
We prove a conjecture on the integrability of Liénard systems
title On a conjecture on the integrability of Liénard systems
spellingShingle On a conjecture on the integrability of Liénard systems
Llibre, Jaume|||0000-0002-9511-5999
Liénard system
First integral
Strong saddle
title_short On a conjecture on the integrability of Liénard systems
title_full On a conjecture on the integrability of Liénard systems
title_fullStr On a conjecture on the integrability of Liénard systems
title_full_unstemmed On a conjecture on the integrability of Liénard systems
title_sort On a conjecture on the integrability of Liénard systems
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Murza, Adrian|||0000-0001-9521-5052
Valls, Clàudia|||0000-0001-8279-1229
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Murza, Adrian|||0000-0001-9521-5052
Valls, Clàudia|||0000-0001-8279-1229
author_role author
author2 Murza, Adrian|||0000-0001-9521-5052
Valls, Clàudia|||0000-0001-8279-1229
author2_role author
author
dc.subject.none.fl_str_mv Liénard system
First integral
Strong saddle
topic Liénard system
First integral
Strong saddle
description 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.
publishDate 2020
dc.date.none.fl_str_mv 2
2020-01-01
2020
2020-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/221375
https://dx.doi.org/urn:doi:10.1007/s12215-018-00398-6
url https://ddd.uab.cat/record/221375
https://dx.doi.org/urn:doi:10.1007/s12215-018-00398-6
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-77278-P
Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2013-40998-P
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-568
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
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eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
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