Parabolic saddles and newhouse domains in celestial mechanics

In McGehee (J Differ Equ 14:70–88, 1973) McGehee introduced a compactification of the phase space of the restricted 3-body problem by gluing a manifold of periodic orbits “at infinity”. Although from the dynamical point of view these periodic orbits are parabolic (the linearization of the Poincaré m...

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Autores: Garrido Peláez, Miguel, Martín de la Torre, Pablo|||0000-0002-0273-1208, Paradela Díaz, Jaime
Tipo de recurso: artículo
Fecha de publicación:2025
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:2117/445899
Acceso en línea:https://hdl.handle.net/2117/445899
https://dx.doi.org/10.1007/s00220-025-05299-1
Access Level:acceso abierto
Palabra clave:Differential Equations
Dynamical Systems
Multistability
Nonlinear Dynamics and Chaos Theory
Ordinary Differential Equations
Partial Differential Equations on Manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
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spelling Parabolic saddles and newhouse domains in celestial mechanicsGarrido Peláez, MiguelMartín de la Torre, Pablo|||0000-0002-0273-1208Paradela Díaz, JaimeDifferential EquationsDynamical SystemsMultistabilityNonlinear Dynamics and Chaos TheoryOrdinary Differential EquationsPartial Differential Equations on ManifoldsÀrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integralsIn McGehee (J Differ Equ 14:70–88, 1973) McGehee introduced a compactification of the phase space of the restricted 3-body problem by gluing a manifold of periodic orbits “at infinity”. Although from the dynamical point of view these periodic orbits are parabolic (the linearization of the Poincaré map is the identity matrix), one of them, denoted here by O, possesses stable and unstable manifolds which, moreover, separate the regions of bounded and unbounded motion. This observation prompted the investigation of the homoclinic picture associated to O, starting with the work of Alekseev (Uspehi Mat Nauk 23:209–210, 1968), Alekseev (Mat Sb 77(119):545–601, 1968) and Moser (Stable and random motions in dynamical systems. Princeton Landmarks in Mathematics. With special emphasis on celestial mechanics, Reprint of the 1973 original, With a foreword by Philip J. Holmes, 2001). We continue this research and extend, to this degenerate setting, some classical results in the theory of homoclinic bifurcations. More concretely, we prove that there exist Newhouse domains N in parameter space (the ratio of masses of the bodies) and residual subsets R ¿ N for which the homoclinic class of O has maximal Hausdorff dimension and is accumulated by generic elliptic periodic orbits. One of the main consequences of our work is the fact that, for a (locally) topologically large set of parameters of the restricted 3-body problem the union of its elliptic islands forms an unbounded subset of the phase space and, moreover, the closure of the set of generic elliptic periodic orbits contains hyperbolic sets with Hausdorff dimension arbitrarily close to maximal. Other instances of the restricted n-body problem such as the Sitnikov problem and the case n = 4 are also considered.J.P. wants to express his gratitude to V. Kaloshin for hintful conversations and valuable suggestions, as well as for sharing with him the unpublished manuscript [GK12]. M.G. has been partially supported by the Spanish Government grants PID2022-136613NB-I00 and PRE2020-096613 and the Catalan Government grant 2021SGR00113. P.M. has been partially supported by the grant PID2021-123968NB-I00, funded by the Spanish State Research Agency through the programs MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe” and by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).Peer Reviewed20252025-07-0120252025-11-10journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/445899https://dx.doi.org/10.1007/s00220-025-05299-1reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2021-123968NB-I00 METODOS MODERNOS EN MECANICA CELESTE Y APLICACIONESAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2022-136613NB-I00 SISTEMAS DINAMICOS CONTINUOS Y DISCRETOS: BIFURCACIONES, ORBITAS PERIODICAS, INTEGRABILIDAD Y APLICACIONESopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4458992026-05-27T15:37:01Z
dc.title.none.fl_str_mv Parabolic saddles and newhouse domains in celestial mechanics
title Parabolic saddles and newhouse domains in celestial mechanics
spellingShingle Parabolic saddles and newhouse domains in celestial mechanics
Garrido Peláez, Miguel
Differential Equations
Dynamical Systems
Multistability
Nonlinear Dynamics and Chaos Theory
Ordinary Differential Equations
Partial Differential Equations on Manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
title_short Parabolic saddles and newhouse domains in celestial mechanics
title_full Parabolic saddles and newhouse domains in celestial mechanics
title_fullStr Parabolic saddles and newhouse domains in celestial mechanics
title_full_unstemmed Parabolic saddles and newhouse domains in celestial mechanics
title_sort Parabolic saddles and newhouse domains in celestial mechanics
dc.creator.none.fl_str_mv Garrido Peláez, Miguel
Martín de la Torre, Pablo|||0000-0002-0273-1208
Paradela Díaz, Jaime
author Garrido Peláez, Miguel
author_facet Garrido Peláez, Miguel
Martín de la Torre, Pablo|||0000-0002-0273-1208
Paradela Díaz, Jaime
author_role author
author2 Martín de la Torre, Pablo|||0000-0002-0273-1208
Paradela Díaz, Jaime
author2_role author
author
dc.subject.none.fl_str_mv Differential Equations
Dynamical Systems
Multistability
Nonlinear Dynamics and Chaos Theory
Ordinary Differential Equations
Partial Differential Equations on Manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
topic Differential Equations
Dynamical Systems
Multistability
Nonlinear Dynamics and Chaos Theory
Ordinary Differential Equations
Partial Differential Equations on Manifolds
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
description In McGehee (J Differ Equ 14:70–88, 1973) McGehee introduced a compactification of the phase space of the restricted 3-body problem by gluing a manifold of periodic orbits “at infinity”. Although from the dynamical point of view these periodic orbits are parabolic (the linearization of the Poincaré map is the identity matrix), one of them, denoted here by O, possesses stable and unstable manifolds which, moreover, separate the regions of bounded and unbounded motion. This observation prompted the investigation of the homoclinic picture associated to O, starting with the work of Alekseev (Uspehi Mat Nauk 23:209–210, 1968), Alekseev (Mat Sb 77(119):545–601, 1968) and Moser (Stable and random motions in dynamical systems. Princeton Landmarks in Mathematics. With special emphasis on celestial mechanics, Reprint of the 1973 original, With a foreword by Philip J. Holmes, 2001). We continue this research and extend, to this degenerate setting, some classical results in the theory of homoclinic bifurcations. More concretely, we prove that there exist Newhouse domains N in parameter space (the ratio of masses of the bodies) and residual subsets R ¿ N for which the homoclinic class of O has maximal Hausdorff dimension and is accumulated by generic elliptic periodic orbits. One of the main consequences of our work is the fact that, for a (locally) topologically large set of parameters of the restricted 3-body problem the union of its elliptic islands forms an unbounded subset of the phase space and, moreover, the closure of the set of generic elliptic periodic orbits contains hyperbolic sets with Hausdorff dimension arbitrarily close to maximal. Other instances of the restricted n-body problem such as the Sitnikov problem and the case n = 4 are also considered.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025-07-01
2025
2025-11-10
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/2117/445899
https://dx.doi.org/10.1007/s00220-025-05299-1
url https://hdl.handle.net/2117/445899
https://dx.doi.org/10.1007/s00220-025-05299-1
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 http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2021-123968NB-I00 METODOS MODERNOS EN MECANICA CELESTE Y APLICACIONES
Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2022-136613NB-I00 SISTEMAS DINAMICOS CONTINUOS Y DISCRETOS: BIFURCACIONES, ORBITAS PERIODICAS, INTEGRABILIDAD Y APLICACIONES
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
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
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
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