{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,8]],"date-time":"2025-01-08T05:19:07Z","timestamp":1736313547480,"version":"3.32.0"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,2]],"date-time":"2021-04-02T00:00:00Z","timestamp":1617321600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Inter-University Centre for Astronomy and Astrophysics (IUCAA)","award":["No number"]},{"DOI":"10.13039\/100007801","name":"Fundacion Seneca","doi-asserted-by":"publisher","award":["20783\/PI\/18"],"id":[{"id":"10.13039\/100007801","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministry of Science, Innovation and Universities","doi-asserted-by":"publisher","award":["PGC2018-097198-B-100"],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Applied Computer Science, Faculty of Electrical, Electronic, Computer and Control Engineering, Lodz University of Technology","award":["501\/2-24-1-2"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The relative motion of an outline of the rendezvous problem has been studied by assuming that the chief satellite is in circular symmetric orbits. The legitimacy of perturbation techniques and nonlinear relative motion are investigated. The deputy satellite equations of motion with respect to the fixed references at the center of the chief satellite are nonlinear in the general case. We found the periodic solutions of the linear relative motion satellite and for the nonlinear relative motion satellite using the Lindstedt\u2013Poincar\u00e9 technique. Comparisons among the analytical solutions of linear and nonlinear motions and the obtained solution by the numerical integration of the explicit Euler method for both motions are investigated. We demonstrate that both analytical and numerical solutions of linear motion are symmetric periodic. However, the solutions of nonlinear motion obtained by the Lindstedt\u2013Poincar\u00e9 technique are periodic and the numerical solutions obtained by integration by using explicit Euler method are non-periodic. Thus, the Lindstedt\u2013Poincar\u00e9 technique is recommended for designing the periodic solutions. Furthermore, a comparison between linear and nonlinear analytical solutions of relative motion is investigated graphically.<\/jats:p>","DOI":"10.3390\/sym13040595","type":"journal-article","created":{"date-parts":[[2021,4,2]],"date-time":"2021-04-02T14:34:09Z","timestamp":1617374049000},"page":"595","source":"Crossref","is-referenced-by-count":10,"title":["Periodic Solutions of Nonlinear Relative Motion Satellites"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4445-1052","authenticated-orcid":false,"given":"Ashok Kumar","family":"Pal","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur 303007, Rajasthan, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2800-4527","authenticated-orcid":false,"given":"Elbaz I.","family":"Abouelmagd","sequence":"additional","affiliation":[{"name":"Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421, Cairo, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan Luis","family":"Garc\u00eda Guirao","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada y Estad\u00edstica, Universidad Polit\u00e9cnica de Cartagena, Hospital de Marina, 30203 Cartagena, Regi\u00f3n de Murcia, Spain"},{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7207-5253","authenticated-orcid":false,"given":"Dariusz W.","family":"Brzezi\u0144ski","sequence":"additional","affiliation":[{"name":"Institute of Applied Computer Science, Faculty of Electrical, Electronic, Computer and Control Engineering, Lodz University of Technology, ul. Stefanowskiego 18\/22, 90-924 \u0141\u00f3d\u017a, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.2307\/2369430","article-title":"Researches in the lunar theory","volume":"1","author":"Hill","year":"1878","journal-title":"Am. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"653","DOI":"10.2514\/8.8704","article-title":"Terminal guidance system for satellite rendezvous","volume":"27","author":"Clohessy","year":"1960","journal-title":"J. Aerosp. 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