{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T18:19:44Z","timestamp":1778782784520,"version":"3.51.4"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2005,8]]},"abstract":"<jats:p> We study numerically the three-dimensional periodic motion of a massless particle in an annular configuration of N bodies. This ring model can approximate the dynamics of celestial systems like clusters and planetary rings. After calculating numerically the families of x-symmetric periodic orbits in the planar case for a certain N, we investigate the families' vertical stability and locate those members that present critical instability in the eigenvalues corresponding to the z-axis. These orbits are the points of bifurcation of the planar families of periodic orbits to three-dimensional families. By means of differentially correcting the vertically critical orbits we extend our numerical investigation in the three-dimensional space and locate several families of three-dimensional periodic orbits symmetric to the xz-plane. The evolution of the families and the significant features of their members are subsequently presented. <\/jats:p>","DOI":"10.1142\/s0218127405013617","type":"journal-article","created":{"date-parts":[[2005,9,29]],"date-time":"2005-09-29T09:34:54Z","timestamp":1127986494000},"page":"2681-2688","source":"Crossref","is-referenced-by-count":22,"title":["NUMERICAL INVESTIGATION OF PERIODIC MOTION IN THE THREE-DIMENSIONAL RING PROBLEM OF N BODIES"],"prefix":"10.1142","volume":"15","author":[{"given":"K. G.","family":"HADJIFOTINOU","sequence":"first","affiliation":[{"name":"Department of Mathematics, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T. J.","family":"KALVOURIDIS","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Applied Sciences, National Technical University of Athens, Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/S0093-6413(03)00086-7"},{"key":"rf2","first-page":"1","volume":"2","author":"Goudas C. L.","journal-title":"Bull. Soc. Math. De Gr\u00e8ce (N. Serie)"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1023\/A:1026475908041"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/S0032-0633(02)00008-9"},{"key":"rf5","first-page":"415","volume":"28","author":"H\u00e9non M.","journal-title":"Astron. Astrophys."},{"key":"rf6","first-page":"30","volume":"81","author":"Ichtiaroglou S.","journal-title":"Astron. Astrophys."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1001739018698"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1023\/A:1002041820281"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011919508410"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(88)80027-0"},{"key":"rf11","first-page":"309","volume":"205","author":"Salo H.","journal-title":"Astron. Astrophys."},{"key":"rf12","volume-title":"Theory of Orbits","author":"Szebehely V.","year":"1967"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127405013617","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T15:04:31Z","timestamp":1565190271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127405013617"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,8]]},"references-count":12,"journal-issue":{"issue":"08","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2005,8]]}},"alternative-id":["10.1142\/S0218127405013617"],"URL":"https:\/\/doi.org\/10.1142\/s0218127405013617","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,8]]}}}