Hopf Bifurcation with Tetrahedral and Octahedral Symmetry
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | https://hdl.handle.net/10216/90682 |
Resumo: | In the study of the periodic solutions of a G-equivariant dynamical system, the H mod K theorem gives all possible periodic solutions, based on group-theoretical aspects. By contrast, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each C-axial subgroup of Gamma x S-1. In this article we compare the bifurcation of periodic solutions for generic differential equations equivariant under the full group of symmetries of the tetrahedron and the group of rotational symmetries of the cube. The two groups are the image of inequivalent representations of the symmetric group S-4. The possible spatial symmetries of bifurcating solutions are different, even though the two groups yield the same group of matrices Gamma x S-1. The same group of matrices occurs again as the extension Gamma x S-1 when G is the full group of symmetries of the cube. For these three groups, while characterizing the Hopf bifurcation, we identify which periodic solution types, whose existence is guaranteed by the H mod K theorem, are obtainable by Hopf bifurcation from the origin. |
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Hopf Bifurcation with Tetrahedral and Octahedral SymmetryIn the study of the periodic solutions of a G-equivariant dynamical system, the H mod K theorem gives all possible periodic solutions, based on group-theoretical aspects. By contrast, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each C-axial subgroup of Gamma x S-1. In this article we compare the bifurcation of periodic solutions for generic differential equations equivariant under the full group of symmetries of the tetrahedron and the group of rotational symmetries of the cube. The two groups are the image of inequivalent representations of the symmetric group S-4. The possible spatial symmetries of bifurcating solutions are different, even though the two groups yield the same group of matrices Gamma x S-1. The same group of matrices occurs again as the extension Gamma x S-1 when G is the full group of symmetries of the cube. For these three groups, while characterizing the Hopf bifurcation, we identify which periodic solution types, whose existence is guaranteed by the H mod K theorem, are obtainable by Hopf bifurcation from the origin.20162016-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/10216/90682eng1536-004010.1137/15m1009317Isabel S LabouriauMurza, ACinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-29T14:27:30Zoai:repositorio-aberto.up.pt:10216/90682Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T00:01:41.411325Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
title |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
spellingShingle |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry Isabel S Labouriau |
title_short |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
title_full |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
title_fullStr |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
title_full_unstemmed |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
title_sort |
Hopf Bifurcation with Tetrahedral and Octahedral Symmetry |
author |
Isabel S Labouriau |
author_facet |
Isabel S Labouriau Murza, AC |
author_role |
author |
author2 |
Murza, AC |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Isabel S Labouriau Murza, AC |
description |
In the study of the periodic solutions of a G-equivariant dynamical system, the H mod K theorem gives all possible periodic solutions, based on group-theoretical aspects. By contrast, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each C-axial subgroup of Gamma x S-1. In this article we compare the bifurcation of periodic solutions for generic differential equations equivariant under the full group of symmetries of the tetrahedron and the group of rotational symmetries of the cube. The two groups are the image of inequivalent representations of the symmetric group S-4. The possible spatial symmetries of bifurcating solutions are different, even though the two groups yield the same group of matrices Gamma x S-1. The same group of matrices occurs again as the extension Gamma x S-1 when G is the full group of symmetries of the cube. For these three groups, while characterizing the Hopf bifurcation, we identify which periodic solution types, whose existence is guaranteed by the H mod K theorem, are obtainable by Hopf bifurcation from the origin. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016 2016-01-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://hdl.handle.net/10216/90682 |
url |
https://hdl.handle.net/10216/90682 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1536-0040 10.1137/15m1009317 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799135940985225216 |