The period function for second-order quadratic ODEs is monotone

Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered in the family of second-order differential equations ¨x = V (x, ˙ x), being V a quadratic polynomial, should have a mo...

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Autores: Gasull Embid, Armengol, Guillamon Grabolosa, Antoni|||0000-0001-8268-4503, Villadelprat Yagüe, Jordi
Tipo de recurso: artículo
Fecha de publicación:2003
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/847
Acceso en línea:https://hdl.handle.net/2117/847
Access Level:acceso abierto
Palabra clave:Differential equations
Differentiable dynamical systems
period function
Equacions diferencials ordinàries
Sistemes dinàmics diferenciables
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
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spelling The period function for second-order quadratic ODEs is monotoneGasull Embid, ArmengolGuillamon Grabolosa, Antoni|||0000-0001-8268-4503Villadelprat Yagüe, JordiDifferential equationsDifferentiable dynamical systemsperiod functionEquacions diferencials ordinàriesSistemes dinàmics diferenciablesClassificació AMS::34 Ordinary differential equations::34C Qualitative theoryClassificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theoryVery little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered in the family of second-order differential equations ¨x = V (x, ˙ x), being V a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [?]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true when V is a polynomial which nonlinear part is homogeneous of degree n > 2.20032003-01-0120072007-05-02journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/847reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/8472026-05-27T15:37:01Z
dc.title.none.fl_str_mv The period function for second-order quadratic ODEs is monotone
title The period function for second-order quadratic ODEs is monotone
spellingShingle The period function for second-order quadratic ODEs is monotone
Gasull Embid, Armengol
Differential equations
Differentiable dynamical systems
period function
Equacions diferencials ordinàries
Sistemes dinàmics diferenciables
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
title_short The period function for second-order quadratic ODEs is monotone
title_full The period function for second-order quadratic ODEs is monotone
title_fullStr The period function for second-order quadratic ODEs is monotone
title_full_unstemmed The period function for second-order quadratic ODEs is monotone
title_sort The period function for second-order quadratic ODEs is monotone
dc.creator.none.fl_str_mv Gasull Embid, Armengol
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Villadelprat Yagüe, Jordi
author Gasull Embid, Armengol
author_facet Gasull Embid, Armengol
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Villadelprat Yagüe, Jordi
author_role author
author2 Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Villadelprat Yagüe, Jordi
author2_role author
author
dc.subject.none.fl_str_mv Differential equations
Differentiable dynamical systems
period function
Equacions diferencials ordinàries
Sistemes dinàmics diferenciables
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
topic Differential equations
Differentiable dynamical systems
period function
Equacions diferencials ordinàries
Sistemes dinàmics diferenciables
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
description Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered in the family of second-order differential equations ¨x = V (x, ˙ x), being V a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [?]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true when V is a polynomial which nonlinear part is homogeneous of degree n > 2.
publishDate 2003
dc.date.none.fl_str_mv 2003
2003-01-01
2007
2007-05-02
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/847
url https://hdl.handle.net/2117/847
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301603

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