{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:29:45Z","timestamp":1787228985292,"version":"build-2736575974"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1220628"],"award-info":[{"award-number":["1220628"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>This work is a continuation of [A. Alberti and C. Vidal, Proc. Roy. Soc. Edinburgh Sect. A Math., 154 (2024), pp. 961\u2013992] where the planar case was considered. Here we investigate the existence of families of symmetric periodic solutions as continuation of the elliptic orbits of the three-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. We consider spatial perturbations of the nonrotating and the rotating Kepler problem that are invariant with respect to some discrete reflection symmetries or have axial symmetry. More precisely, we characterize the sufficient conditions for their existence, and their type of stability is studied by estimating the characteristic multipliers of the symmetric periodic solutions. In addition, we present some results which establish a relation between symmetric periodic solutions obtained by the analytical continuation method and those obtained by the averaging method for 3-DOF Hamiltonian systems. Our study first takes into account that in each method it is necessary to find points from which the initial conditions that generating the families of periodic solutions bifurcate. By imposing a discrete symmetry on the average function, we can relate the previous points. Second, we show that the non-degeneracy condition for the application of the averaging method implies the non-degeneracy condition of the Poincar\u00e9 continuation method. Finally, we compare the periods and characteristic multipliers of the family of periodic solutions obtained by each method. New families of periodic solutions for the generalized hydrogen atom in external electric and magnetic fields and for the hydrogen atom in van der Waals potential combined with a magnetic field are obtained as an application of our main results.<\/jats:p>","DOI":"10.1137\/24m1666689","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T04:02:16Z","timestamp":1741579336000},"page":"782-814","source":"Crossref","is-referenced-by-count":2,"title":["Spatial Near-Elliptical Symmetric Periodic Orbits of Perturbed Kepler Problems: From Existence to Stability"],"prefix":"10.1137","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5991-4208","authenticated-orcid":true,"given":"Angelo","family":"Alberti","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Federal de Sergipe, Cidade universit\u00e1ria Prof. Jos\u00e9 Alo\u00edsio de Campos, Rosa Elze, S\u00e3o Crist\u00f3v\u00e3o-SE, Brasil."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1630-0898","authenticated-orcid":true,"given":"Claudio","family":"Vidal","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ciencias, Universidad del B\u00edo-B\u00edo, Concepci\u00f3n, VIII Regi\u00f3n, Chile."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,3,10]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127417500390"},{"key":"ref2","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.3934\/dcdsb.2019003","volume":"24","author":"Abouelmagd E. 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