Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a Hamiltonian depending on one parameter, K¿>¿0. The corresponding Hamiltonian system of ODE has two equilibrium points L1 (unstable for all K and energy value h(L1)) and L2 (a center for K¿...

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Autores: Ollé Torner, Mercè|||0000-0002-8050-9055, Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
Tipo de recurso: artículo
Fecha de publicación:2018
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/118546
Acceso en línea:https://hdl.handle.net/2117/118546
https://dx.doi.org/10.1016/j.cnsns.2018.02.005
Access Level:acceso abierto
Palabra clave:Hamiltonian systems
Invariant manifolds
Hamiltonian dynamical systems
Hopf bifurcation
Periodic orbits and tori
Chaotic regions
Àrees temàtiques de la UPC::Matemàtiques i estadística
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spelling Hopf bifurcation for the hydrogen atom in a circularly polarized microwave fieldOllé Torner, Mercè|||0000-0002-8050-9055Pacha Andújar, Juan Ramón|||0000-0003-4599-3141Hamiltonian systemsInvariant manifoldsHamiltonian dynamical systemsHopf bifurcationPeriodic orbits and toriChaotic regionsInvariant manifoldsÀrees temàtiques de la UPC::Matemàtiques i estadísticaWe consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a Hamiltonian depending on one parameter, K¿>¿0. The corresponding Hamiltonian system of ODE has two equilibrium points L1 (unstable for all K and energy value h(L1)) and L2 (a center for K¿<¿Kcrit and a complex saddle for K¿>¿Kcrit, with energy value h(L2)). We study the Hamiltonian-Hopf bifurcation phenomena that take place for K close to Kcrit around L2. First, a local analysis based on the computation of the integrable normal form up to a finite order is carried out and the steps for the computation of this (resonant) normal form are explained in a constructive manner. The analysis of the normal form obtained allows: to claim the type of the Hopf bifurcation –supercritical–; to study the local behavior of the electron in a neighborhood of the equilibrium L2 for the original non integrable Hamiltonian (as a perturbative approach from the integrable normal form); to obtain (approximations for) the parametrizations of the relevant invariant objects that take place due to the bifurcation (periodic orbits and invariant manifolds of L2). We compute numerically such objects and analyse not only the local picture of the dynamics close to L2, but also a global description of the dynamics and the effect of the Hopf bifurcation as well as other objects that organize the dynamics are discussed. We conclude that, for K close to Kcrit and the energy level h(L2), the Hopf bifurcation has essentially no effect on the dynamics from a physical point of view. However, for bigger values of K¿>¿Kcrit, the Hopf bifurcation has a dramatic effect: different kind of orbits coexist, mostly chaotic. Such orbits provide a ionization mechanism with several passages far from and close to L2 before ionizing. Surprisingly enough, also robust confinement regions (where the electron remains confined for ever), exist in the middle of chaotic areasPeer Reviewed20182018-09-0120182018-06-27journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/118546https://dx.doi.org/10.1016/j.cnsns.2018.02.005reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1185462026-05-27T15:37:01Z
dc.title.none.fl_str_mv Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
title Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
spellingShingle Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
Ollé Torner, Mercè|||0000-0002-8050-9055
Hamiltonian systems
Invariant manifolds
Hamiltonian dynamical systems
Hopf bifurcation
Periodic orbits and tori
Chaotic regions
Invariant manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
title_full Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
title_fullStr Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
title_full_unstemmed Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
title_sort Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field
dc.creator.none.fl_str_mv Ollé Torner, Mercè|||0000-0002-8050-9055
Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
author Ollé Torner, Mercè|||0000-0002-8050-9055
author_facet Ollé Torner, Mercè|||0000-0002-8050-9055
Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
author_role author
author2 Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
author2_role author
dc.subject.none.fl_str_mv Hamiltonian systems
Invariant manifolds
Hamiltonian dynamical systems
Hopf bifurcation
Periodic orbits and tori
Chaotic regions
Invariant manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Hamiltonian systems
Invariant manifolds
Hamiltonian dynamical systems
Hopf bifurcation
Periodic orbits and tori
Chaotic regions
Invariant manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística
description We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a Hamiltonian depending on one parameter, K¿>¿0. The corresponding Hamiltonian system of ODE has two equilibrium points L1 (unstable for all K and energy value h(L1)) and L2 (a center for K¿<¿Kcrit and a complex saddle for K¿>¿Kcrit, with energy value h(L2)). We study the Hamiltonian-Hopf bifurcation phenomena that take place for K close to Kcrit around L2. First, a local analysis based on the computation of the integrable normal form up to a finite order is carried out and the steps for the computation of this (resonant) normal form are explained in a constructive manner. The analysis of the normal form obtained allows: to claim the type of the Hopf bifurcation –supercritical–; to study the local behavior of the electron in a neighborhood of the equilibrium L2 for the original non integrable Hamiltonian (as a perturbative approach from the integrable normal form); to obtain (approximations for) the parametrizations of the relevant invariant objects that take place due to the bifurcation (periodic orbits and invariant manifolds of L2). We compute numerically such objects and analyse not only the local picture of the dynamics close to L2, but also a global description of the dynamics and the effect of the Hopf bifurcation as well as other objects that organize the dynamics are discussed. We conclude that, for K close to Kcrit and the energy level h(L2), the Hopf bifurcation has essentially no effect on the dynamics from a physical point of view. However, for bigger values of K¿>¿Kcrit, the Hopf bifurcation has a dramatic effect: different kind of orbits coexist, mostly chaotic. Such orbits provide a ionization mechanism with several passages far from and close to L2 before ionizing. Surprisingly enough, also robust confinement regions (where the electron remains confined for ever), exist in the middle of chaotic areas
publishDate 2018
dc.date.none.fl_str_mv 2018
2018-09-01
2018
2018-06-27
dc.type.none.fl_str_mv journal 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://hdl.handle.net/2117/118546
https://dx.doi.org/10.1016/j.cnsns.2018.02.005
url https://hdl.handle.net/2117/118546
https://dx.doi.org/10.1016/j.cnsns.2018.02.005
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
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
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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