Parabolic Saddles and Newhouse Domains in Celestial Mechanics
Garrido, Miguel (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Martín, Pau (Universitat Politècnica de Catalunya. Departament de Matemàtiques)
Paradela, Jaime (University of Maryland. Department of Mathematics)

Data:	2025
Resum:	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.
Ajuts:	Agencia Estatal de Investigación PID2022-136613NB-I00 Agencia Estatal de Investigación PRE2020-096613 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113 Agencia Estatal de Investigación PID2021-123968NB-I00 Agencia Estatal de Investigación CEX2020-001084-M
Nota:	Altres ajuts: acords transformatius de la UAB
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Publicat a:	Communications in mathematical physics, Vol. 406 (June 2025) , art. 173, ISSN 1432-0916

DOI: 10.1007/s00220-025-05299-1

67 p, 999.8 KB

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