Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation

We consider a one parameter family of 2-DOF Hamiltonian systems having an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation. We briefly review the well-established normal form theory in this case. Then we focus on the invariant manifolds when there are homoclinic orbits to the complex-...

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Detalhes bibliográficos
Autores: Fontich, Ernest, 1955-, Vieiro Yanes, Arturo
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2022
País:España
Recursos:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/191730
Acesso em linha:https://hdl.handle.net/2445/191730
Access Level:acceso abierto
Palavra-chave:Sistemes hamiltonians
Sistemes dinàmics diferenciables
Varietats (Matemàtica)
Hamiltonian systems
Differentiable dynamical systems
Manifolds (Mathematics)
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spelling Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcationFontich, Ernest, 1955-Vieiro Yanes, ArturoSistemes hamiltoniansSistemes dinàmics diferenciablesVarietats (Matemàtica)Hamiltonian systemsDifferentiable dynamical systemsManifolds (Mathematics)We consider a one parameter family of 2-DOF Hamiltonian systems having an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation. We briefly review the well-established normal form theory in this case. Then we focus on the invariant manifolds when there are homoclinic orbits to the complex-saddle equilibrium point, and we study the behavior of the splitting of the 2D invariant manifolds. The symmetries of the normal form are used to reduce the dynamics around the invariant manifolds to the dynamics of a family of area-preserving near-identity Poincaré maps that can be extended analytically to a suitable neighborhood of the separatrices. This allows, in particular, to use well-known results for area-preserving maps and derive an explicit upper bound of the splitting of separatrices for the Poincaré map. We illustrate the results in a concrete example. Different Poincaré sections are used to visualize the dynamics near the 2D invariant manifolds. Last section deals with the derivation of a separatrix map to study the chaotic dynamics near the 2D invariant manifolds.Elsevier B.V.2022202220232022info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/191730Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció del document publicat a: https://doi.org/10.1016/j.cnsns.2022.106971Communications In Nonlinear Science And Numerical Simulation, 2023, vol. 117, p. 106971https://doi.org/10.1016/j.cnsns.2022.106971cc by-nc-nd (c) Ernest Fontich et al., 2023http://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:recercat.cat:2445/1917302026-05-29T05:05:01Z
dc.title.none.fl_str_mv Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
title Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
spellingShingle Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
Fontich, Ernest, 1955-
Sistemes hamiltonians
Sistemes dinàmics diferenciables
Varietats (Matemàtica)
Hamiltonian systems
Differentiable dynamical systems
Manifolds (Mathematics)
title_short Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
title_full Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
title_fullStr Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
title_full_unstemmed Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
title_sort Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation
dc.creator.none.fl_str_mv Fontich, Ernest, 1955-
Vieiro Yanes, Arturo
author Fontich, Ernest, 1955-
author_facet Fontich, Ernest, 1955-
Vieiro Yanes, Arturo
author_role author
author2 Vieiro Yanes, Arturo
author2_role author
dc.subject.none.fl_str_mv Sistemes hamiltonians
Sistemes dinàmics diferenciables
Varietats (Matemàtica)
Hamiltonian systems
Differentiable dynamical systems
Manifolds (Mathematics)
topic Sistemes hamiltonians
Sistemes dinàmics diferenciables
Varietats (Matemàtica)
Hamiltonian systems
Differentiable dynamical systems
Manifolds (Mathematics)
description We consider a one parameter family of 2-DOF Hamiltonian systems having an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation. We briefly review the well-established normal form theory in this case. Then we focus on the invariant manifolds when there are homoclinic orbits to the complex-saddle equilibrium point, and we study the behavior of the splitting of the 2D invariant manifolds. The symmetries of the normal form are used to reduce the dynamics around the invariant manifolds to the dynamics of a family of area-preserving near-identity Poincaré maps that can be extended analytically to a suitable neighborhood of the separatrices. This allows, in particular, to use well-known results for area-preserving maps and derive an explicit upper bound of the splitting of separatrices for the Poincaré map. We illustrate the results in a concrete example. Different Poincaré sections are used to visualize the dynamics near the 2D invariant manifolds. Last section deals with the derivation of a separatrix map to study the chaotic dynamics near the 2D invariant manifolds.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022
2022
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/191730
url https://hdl.handle.net/2445/191730
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.1016/j.cnsns.2022.106971
Communications In Nonlinear Science And Numerical Simulation, 2023, vol. 117, p. 106971
https://doi.org/10.1016/j.cnsns.2022.106971
dc.rights.none.fl_str_mv cc by-nc-nd (c) Ernest Fontich et al., 2023
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc by-nc-nd (c) Ernest Fontich et al., 2023
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier B.V.
publisher.none.fl_str_mv Elsevier B.V.
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,81155

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