Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems

In the R2 plane, the simplest non-linear differential systems are the quadratic polynomial differential systems. This type of differential systems have been studied intensively because of its non-linearity and wide range of applications. These systems have been classified into ten classes. In this a...

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Autores: Artés Ferragud, Joan Carles|||0000-0003-4332-7495, Cairó, Laurent, Llibre, Jaume|||0000-0002-9511-5999
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:311941
Acceso en línea:https://ddd.uab.cat/record/311941
https://dx.doi.org/urn:doi:10.1007/s12346-025-01227-9
Access Level:acceso abierto
Palabra clave:Phase portrait
Quadratic system
Quadratic vector field
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spelling Phase Portraits of the Family IV of the Quadratic Polynomial Differential SystemsArtés Ferragud, Joan Carles|||0000-0003-4332-7495Cairó, LaurentLlibre, Jaume|||0000-0002-9511-5999Phase portraitQuadratic systemQuadratic vector fieldIn the R2 plane, the simplest non-linear differential systems are the quadratic polynomial differential systems. This type of differential systems have been studied intensively because of its non-linearity and wide range of applications. These systems have been classified into ten classes. In this article, we characterize in the Poincarè disk all topologically different phase portraits for one of these classes. Concretely we provide the complete study of the geometry of family IV. The family IV is six-dimensional and is reduced to several subfamilies which are three-dimensional. We give the bifurcation diagram of each specific normal form using invariant polynomials. The split of all quadratic systems in ten subfamilies could have been a good way to find all possible phase portraits of quadratic systems, if all ten families could have been studied completely. All families are six-dimensional and some of them can even be divided in several subfamilies with fewer parameters. However, certain families (I, II, and III) are too generic and cannot be reduced enough to allow its complete study. Regardless, this division of quadratic systems has been useful to study several families of quadratic systems having some other properties, and it is worth trying to get as many of them as possible studied in a complete way. 22025-01-0120252025-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/311941https://dx.doi.org/urn:doi:10.1007/s12346-025-01227-9reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-136613NB-I00Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00113open accesshttp://purl.org/coar/access_right/c_abf2Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:3119412026-06-06T12:50:31Z
dc.title.none.fl_str_mv Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
title Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
spellingShingle Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
Artés Ferragud, Joan Carles|||0000-0003-4332-7495
Phase portrait
Quadratic system
Quadratic vector field
title_short Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
title_full Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
title_fullStr Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
title_full_unstemmed Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
title_sort Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems
dc.creator.none.fl_str_mv Artés Ferragud, Joan Carles|||0000-0003-4332-7495
Cairó, Laurent
Llibre, Jaume|||0000-0002-9511-5999
author Artés Ferragud, Joan Carles|||0000-0003-4332-7495
author_facet Artés Ferragud, Joan Carles|||0000-0003-4332-7495
Cairó, Laurent
Llibre, Jaume|||0000-0002-9511-5999
author_role author
author2 Cairó, Laurent
Llibre, Jaume|||0000-0002-9511-5999
author2_role author
author
dc.subject.none.fl_str_mv Phase portrait
Quadratic system
Quadratic vector field
topic Phase portrait
Quadratic system
Quadratic vector field
description In the R2 plane, the simplest non-linear differential systems are the quadratic polynomial differential systems. This type of differential systems have been studied intensively because of its non-linearity and wide range of applications. These systems have been classified into ten classes. In this article, we characterize in the Poincarè disk all topologically different phase portraits for one of these classes. Concretely we provide the complete study of the geometry of family IV. The family IV is six-dimensional and is reduced to several subfamilies which are three-dimensional. We give the bifurcation diagram of each specific normal form using invariant polynomials. The split of all quadratic systems in ten subfamilies could have been a good way to find all possible phase portraits of quadratic systems, if all ten families could have been studied completely. All families are six-dimensional and some of them can even be divided in several subfamilies with fewer parameters. However, certain families (I, II, and III) are too generic and cannot be reduced enough to allow its complete study. Regardless, this division of quadratic systems has been useful to study several families of quadratic systems having some other properties, and it is worth trying to get as many of them as possible studied in a complete way.
publishDate 2025
dc.date.none.fl_str_mv 2
2025-01-01
2025
2025-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/311941
https://dx.doi.org/urn:doi:10.1007/s12346-025-01227-9
url https://ddd.uab.cat/record/311941
https://dx.doi.org/urn:doi:10.1007/s12346-025-01227-9
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-136613NB-I00
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00113
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by/4.0/
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://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
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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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collection Dipòsit Digital de Documents de la UAB
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