An explicit expression of the first Liapunov and period constants with applications

In this paper, we study systems in the plane having a critical point with pure imaginary eigenvalues, and we search for effective conditions to discern whether this critical point is a focus or a center; in the case of it being a center, we look for additional conditions in order to be isochronous....

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Detalles Bibliográficos
Autores: Gasull Embid, Armengol, Guillamon Grabolosa, Antoni|||0000-0001-8268-4503, Mañosa Fernández, Víctor|||0000-0002-5082-3334
Tipo de recurso: artículo
Fecha de publicación:1996
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:2117/883
Acceso en línea:https://hdl.handle.net/2117/883
Access Level:acceso abierto
Palabra clave:Differential equations
Center
Isochronous center
Liapunov constant
Period constant
Equacions diferencials ordinàries
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
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spelling An explicit expression of the first Liapunov and period constants with applicationsGasull Embid, ArmengolGuillamon Grabolosa, Antoni|||0000-0001-8268-4503Mañosa Fernández, Víctor|||0000-0002-5082-3334Differential equationsCenterIsochronous centerLiapunov constantPeriod constantEquacions diferencials ordinàriesClassificació AMS::34 Ordinary differential equations::34C Qualitative theoryIn this paper, we study systems in the plane having a critical point with pure imaginary eigenvalues, and we search for effective conditions to discern whether this critical point is a focus or a center; in the case of it being a center, we look for additional conditions in order to be isochronous. We wish to stress that the essential differences between the techniques used in this work and the more usual ones are basically two: the elimination of the integration constants when we consider primitives of functions (see also Remark 3.2) and the fact that we maintain the complex notation in the whole study. Thanks to these aspects, we reach with relative ease an expression of the first three Liapunov constants, $v_3$, $v_5$ and $v_7$, and of the first two period ones, $p_2$ and $p_4$, for a general system. As far as we know, this is the first time that a general and compact expression of $v_7$ has been given. Moreover, the use of a computer algebra system is only needed in the computation of $v_7$ and $p_4$. These results are applied to give a classification of centers and isochronous centers for certain families of differential equations.19961996-01-0120072007-05-04journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/883reponame: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:2117/8832026-05-27T15:37:01Z
dc.title.none.fl_str_mv An explicit expression of the first Liapunov and period constants with applications
title An explicit expression of the first Liapunov and period constants with applications
spellingShingle An explicit expression of the first Liapunov and period constants with applications
Gasull Embid, Armengol
Differential equations
Center
Isochronous center
Liapunov constant
Period constant
Equacions diferencials ordinàries
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
title_short An explicit expression of the first Liapunov and period constants with applications
title_full An explicit expression of the first Liapunov and period constants with applications
title_fullStr An explicit expression of the first Liapunov and period constants with applications
title_full_unstemmed An explicit expression of the first Liapunov and period constants with applications
title_sort An explicit expression of the first Liapunov and period constants with applications
dc.creator.none.fl_str_mv Gasull Embid, Armengol
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Mañosa Fernández, Víctor|||0000-0002-5082-3334
author Gasull Embid, Armengol
author_facet Gasull Embid, Armengol
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Mañosa Fernández, Víctor|||0000-0002-5082-3334
author_role author
author2 Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Mañosa Fernández, Víctor|||0000-0002-5082-3334
author2_role author
author
dc.subject.none.fl_str_mv Differential equations
Center
Isochronous center
Liapunov constant
Period constant
Equacions diferencials ordinàries
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
topic Differential equations
Center
Isochronous center
Liapunov constant
Period constant
Equacions diferencials ordinàries
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
description In this paper, we study systems in the plane having a critical point with pure imaginary eigenvalues, and we search for effective conditions to discern whether this critical point is a focus or a center; in the case of it being a center, we look for additional conditions in order to be isochronous. We wish to stress that the essential differences between the techniques used in this work and the more usual ones are basically two: the elimination of the integration constants when we consider primitives of functions (see also Remark 3.2) and the fact that we maintain the complex notation in the whole study. Thanks to these aspects, we reach with relative ease an expression of the first three Liapunov constants, $v_3$, $v_5$ and $v_7$, and of the first two period ones, $p_2$ and $p_4$, for a general system. As far as we know, this is the first time that a general and compact expression of $v_7$ has been given. Moreover, the use of a computer algebra system is only needed in the computation of $v_7$ and $p_4$. These results are applied to give a classification of centers and isochronous centers for certain families of differential equations.
publishDate 1996
dc.date.none.fl_str_mv 1996
1996-01-01
2007
2007-05-04
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/2117/883
url https://hdl.handle.net/2117/883
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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