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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| Format: | Article |
| Publication date: | 2025 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/445899 |
| Online Access: | https://hdl.handle.net/2117/445899 https://dx.doi.org/10.1007/s00220-025-05299-1 |
| Access Level: | Open access |
| Keyword: | 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 |
| _version_ | 1878731832927715329 |
|---|---|
| author | Garrido Peláez, Miguel |
| author2 | Martín de la Torre, Pablo|||0000-0002-0273-1208 Paradela Díaz, Jaime |
| author2_role | author author |
| author_browse | Garrido Peláez, Miguel Martín de la Torre, Pablo|||0000-0002-0273-1208 Paradela Díaz, Jaime |
| author_facet | Garrido Peláez, Miguel Martín de la Torre, Pablo|||0000-0002-0273-1208 Paradela Díaz, Jaime |
| author_role | author |
| collection | UPCommons. Portal del coneixement obert de la UPC |
| 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. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_377f5e3ece4e28efaf0ffb35facbc58f |
| institution | Universitat Politècnica de Catalunya (UPC) |
| instname_str | Universitat Politècnica de Catalunya (UPC) |
| language | eng |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:upcommons.upc.edu:2117/445899 |
| publishDate | 2025 |
| publishDateSort | 2025 |
| reponame_str | UPCommons. Portal del coneixement obert de la UPC |
| spelling | Parabolic saddles and newhouse domains in celestial mechanics Garrido Peláez, Miguel Martín de la Torre, Pablo|||0000-0002-0273-1208 Paradela Díaz, Jaime 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 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. 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 Reviewed https://hdl.handle.net/2117/445899 https://dx.doi.org/10.1007/s00220-025-05299-1 |
| 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 | 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_short | Parabolic saddles and newhouse domains in celestial mechanics |
| title_sort | Parabolic saddles and newhouse domains in celestial mechanics |
| 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 |
| topic_facet | 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 |
| url | https://hdl.handle.net/2117/445899 https://dx.doi.org/10.1007/s00220-025-05299-1 |