The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2

In 1960 Markus and Yamabe made the conjecture that if a C1 differential system x˙=F(x) in Rn has a unique equilibrium point and DF(x) is Hurwitz for all x∈Rn, then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true in R2 and false...

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Autores: Llibre, Jaume|||0000-0002-9511-5999, Valls, Clàudia|||0000-0001-8279-1229
Tipo de documento: artigo
Data de publicação:2023
País:España
Recursos:Universitat Autònoma de Barcelona
Repositório:Dipòsit Digital de Documents de la UAB
Idioma:inglês
OAI Identifier:oai:ddd.uab.cat:257120
Acesso em linha:https://ddd.uab.cat/record/257120
https://dx.doi.org/urn:doi:10.1007/s10884-021-10110-5
Access Level:Acceso aberto
Palavra-chave:Markus-Yamabe conjecture
Hurwitz matrix
Discontinuous piecewise linear differential systems
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spelling The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2Llibre, Jaume|||0000-0002-9511-5999Valls, Clàudia|||0000-0001-8279-1229Markus-Yamabe conjectureHurwitz matrixDiscontinuous piecewise linear differential systemsIn 1960 Markus and Yamabe made the conjecture that if a C1 differential system x˙=F(x) in Rn has a unique equilibrium point and DF(x) is Hurwitz for all x∈Rn, then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true in R2 and false in Rn for all n≥3. In (The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems, 2020) the authors extended the Markus-Yamabe conjecture to continuous and discontinuous piecewise linear differential systems in Rn separated by a hyperplane, they proved for the continuous systems that the extended conjecture is true in R2 and false in Rn for all n≥3, but for discontinuous systems they proved that the conjecture is false in Rn for all n≥2. In this paper first we show that there are no continuous piecewise linear differential systems separated by a conic×Rn-2 except the linear differential systems in Rn. And after we prove that the extended Markus-Yamabe conjecture to discontinuous piecewise linear differential systems in Rn separated by a conic×Rn-2 is false in Rn for all n≥2. 22023-01-0120232023-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/257120https://dx.doi.org/urn:doi:10.1007/s10884-021-10110-5reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00European Commission https://doi.org/10.13039/501100000780 777911open 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:2571202026-06-06T12:50:31Z
dc.title.none.fl_str_mv The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
title The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
spellingShingle The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
Llibre, Jaume|||0000-0002-9511-5999
Markus-Yamabe conjecture
Hurwitz matrix
Discontinuous piecewise linear differential systems
title_short The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
title_full The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
title_fullStr The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
title_full_unstemmed The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
title_sort The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Valls, Clàudia|||0000-0001-8279-1229
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Valls, Clàudia|||0000-0001-8279-1229
author_role author
author2 Valls, Clàudia|||0000-0001-8279-1229
author2_role author
dc.subject.none.fl_str_mv Markus-Yamabe conjecture
Hurwitz matrix
Discontinuous piecewise linear differential systems
topic Markus-Yamabe conjecture
Hurwitz matrix
Discontinuous piecewise linear differential systems
description In 1960 Markus and Yamabe made the conjecture that if a C1 differential system x˙=F(x) in Rn has a unique equilibrium point and DF(x) is Hurwitz for all x∈Rn, then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true in R2 and false in Rn for all n≥3. In (The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems, 2020) the authors extended the Markus-Yamabe conjecture to continuous and discontinuous piecewise linear differential systems in Rn separated by a hyperplane, they proved for the continuous systems that the extended conjecture is true in R2 and false in Rn for all n≥3, but for discontinuous systems they proved that the conjecture is false in Rn for all n≥2. In this paper first we show that there are no continuous piecewise linear differential systems separated by a conic×Rn-2 except the linear differential systems in Rn. And after we prove that the extended Markus-Yamabe conjecture to discontinuous piecewise linear differential systems in Rn separated by a conic×Rn-2 is false in Rn for all n≥2.
publishDate 2023
dc.date.none.fl_str_mv 2
2023-01-01
2023
2023-01-01
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
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dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/257120
https://dx.doi.org/urn:doi:10.1007/s10884-021-10110-5
url https://ddd.uab.cat/record/257120
https://dx.doi.org/urn:doi:10.1007/s10884-021-10110-5
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 PID2019-104658GB-I00
European Commission https://doi.org/10.13039/501100000780 777911
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
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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
reponame_str Dipòsit Digital de Documents de la UAB
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