A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results

Autores
Revel, Gustavo; Alonso, Diego; Moiola, Jorge Luis
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper a degenerate case of a 2:3 resonant Hopf–Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two- and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and Neimark– Sacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F) bubble interacting with curves of fold-Neimark–Sacker (FNS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R(2) 1:1), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F, FNS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf–Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.
Fil: Revel, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación en Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; Argentina
Fil: Alonso, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación En Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; Argentina
Fil: Moiola, Jorge Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación en Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; Argentina
Materia
Hopf-Hopf Bifurcation
Strong Resonances
Nonlinear Autonomous Oscillators
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/11837
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/11837
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network_name_str CONICET Digital (CONICET)
spelling A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global ResultsRevel, GustavoAlonso, DiegoMoiola, Jorge LuisHopf-Hopf BifurcationStrong ResonancesNonlinear Autonomous Oscillatorshttps://purl.org/becyt/ford/2.2https://purl.org/becyt/ford/2In this paper a degenerate case of a 2:3 resonant Hopf–Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two- and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and Neimark– Sacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F) bubble interacting with curves of fold-Neimark–Sacker (FNS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R(2) 1:1), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F, FNS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf–Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.Fil: Revel, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación en Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; ArgentinaFil: Alonso, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación En Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; ArgentinaFil: Moiola, Jorge Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación en Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; ArgentinaSiam Publications2015-07info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/11837Revel, Gustavo; Alonso, Diego; Moiola, Jorge Luis; A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results; Siam Publications; Siam Journal On Applied Dynamical Systems; 14; 2; 7-2015; 1130-11641536-0040enginfo:eu-repo/semantics/altIdentifier/url/http://epubs.siam.org/doi/10.1137/140968197info:eu-repo/semantics/altIdentifier/doi/10.1137/140968197info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-11-26T08:52:57Zoai:ri.conicet.gov.ar:11336/11837instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-11-26 08:52:57.687CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
title A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
spellingShingle A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
Revel, Gustavo
Hopf-Hopf Bifurcation
Strong Resonances
Nonlinear Autonomous Oscillators
title_short A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
title_full A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
title_fullStr A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
title_full_unstemmed A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
title_sort A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results
dc.creator.none.fl_str_mv Revel, Gustavo
Alonso, Diego
Moiola, Jorge Luis
author Revel, Gustavo
author_facet Revel, Gustavo
Alonso, Diego
Moiola, Jorge Luis
author_role author
author2 Alonso, Diego
Moiola, Jorge Luis
author2_role author
author
dc.subject.none.fl_str_mv Hopf-Hopf Bifurcation
Strong Resonances
Nonlinear Autonomous Oscillators
topic Hopf-Hopf Bifurcation
Strong Resonances
Nonlinear Autonomous Oscillators
purl_subject.fl_str_mv https://purl.org/becyt/ford/2.2
https://purl.org/becyt/ford/2
dc.description.none.fl_txt_mv In this paper a degenerate case of a 2:3 resonant Hopf–Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two- and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and Neimark– Sacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F) bubble interacting with curves of fold-Neimark–Sacker (FNS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R(2) 1:1), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F, FNS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf–Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.
Fil: Revel, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación en Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; Argentina
Fil: Alonso, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación En Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; Argentina
Fil: Moiola, Jorge Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Investigación en Ingeniería Eléctrica; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria Electrica y de Computadoras; Argentina
description In this paper a degenerate case of a 2:3 resonant Hopf–Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two- and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and Neimark– Sacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F) bubble interacting with curves of fold-Neimark–Sacker (FNS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R(2) 1:1), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F, FNS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf–Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.
publishDate 2015
dc.date.none.fl_str_mv 2015-07
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/11837
Revel, Gustavo; Alonso, Diego; Moiola, Jorge Luis; A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results; Siam Publications; Siam Journal On Applied Dynamical Systems; 14; 2; 7-2015; 1130-1164
1536-0040
url http://hdl.handle.net/11336/11837
identifier_str_mv Revel, Gustavo; Alonso, Diego; Moiola, Jorge Luis; A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results; Siam Publications; Siam Journal On Applied Dynamical Systems; 14; 2; 7-2015; 1130-1164
1536-0040
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://epubs.siam.org/doi/10.1137/140968197
info:eu-repo/semantics/altIdentifier/doi/10.1137/140968197
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Siam Publications
publisher.none.fl_str_mv Siam Publications
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.011256

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