Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x' = -yG(x, y), y' = xG(x, y), where the set of critical points {G(x, y) = 0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study...

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Autores: Gasull, Armengol|||0000-0002-1719-8231, Torregrosa, Joan|||0000-0002-2753-1827
Tipo de recurso: artículo
Fecha de publicación:2012
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:150551
Acceso en línea:https://ddd.uab.cat/record/150551
https://dx.doi.org/urn:doi:10.1016/j.na.2012.04.033
Access Level:acceso abierto
Palabra clave:Abelian integrals
Weak 16th Hilbert's Problem
Limit cycles
Chebyshev system
Number of zeroes of real functions
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spelling Upper bounds for the number of zeroes for some Abelian integralsGasull, Armengol|||0000-0002-1719-8231Torregrosa, Joan|||0000-0002-2753-1827Abelian integralsWeak 16th Hilbert's ProblemLimit cyclesChebyshev systemNumber of zeroes of real functionsConsider the vector field x' = -yG(x, y), y' = xG(x, y), where the set of critical points {G(x, y) = 0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K ≤ 4 we recover or improve some results obtained in several previous works. 22012-01-0120122012-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/150551https://dx.doi.org/urn:doi:10.1016/j.na.2012.04.033reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-410Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2009-06973Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2009/SGR-859open 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:1505512026-06-06T12:50:31Z
dc.title.none.fl_str_mv Upper bounds for the number of zeroes for some Abelian integrals
title Upper bounds for the number of zeroes for some Abelian integrals
spellingShingle Upper bounds for the number of zeroes for some Abelian integrals
Gasull, Armengol|||0000-0002-1719-8231
Abelian integrals
Weak 16th Hilbert's Problem
Limit cycles
Chebyshev system
Number of zeroes of real functions
title_short Upper bounds for the number of zeroes for some Abelian integrals
title_full Upper bounds for the number of zeroes for some Abelian integrals
title_fullStr Upper bounds for the number of zeroes for some Abelian integrals
title_full_unstemmed Upper bounds for the number of zeroes for some Abelian integrals
title_sort Upper bounds for the number of zeroes for some Abelian integrals
dc.creator.none.fl_str_mv Gasull, Armengol|||0000-0002-1719-8231
Torregrosa, Joan|||0000-0002-2753-1827
author Gasull, Armengol|||0000-0002-1719-8231
author_facet Gasull, Armengol|||0000-0002-1719-8231
Torregrosa, Joan|||0000-0002-2753-1827
author_role author
author2 Torregrosa, Joan|||0000-0002-2753-1827
author2_role author
dc.subject.none.fl_str_mv Abelian integrals
Weak 16th Hilbert's Problem
Limit cycles
Chebyshev system
Number of zeroes of real functions
topic Abelian integrals
Weak 16th Hilbert's Problem
Limit cycles
Chebyshev system
Number of zeroes of real functions
description Consider the vector field x' = -yG(x, y), y' = xG(x, y), where the set of critical points {G(x, y) = 0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K ≤ 4 we recover or improve some results obtained in several previous works.
publishDate 2012
dc.date.none.fl_str_mv 2
2012-01-01
2012
2012-01-01
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/150551
https://dx.doi.org/urn:doi:10.1016/j.na.2012.04.033
url https://ddd.uab.cat/record/150551
https://dx.doi.org/urn:doi:10.1016/j.na.2012.04.033
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-410
Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2009-06973
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2009/SGR-859
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
https://rightsstatements.org/vocab/InC/1.0/
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
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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