{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:34:44Z","timestamp":1760243684762,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T00:00:00Z","timestamp":1665100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"S\u00e3o Paulo Research Foundation","award":["2020\/02415-7","2018\/14685-9","2021\/09519-5","2019\/14038-6","2017\/14414-2","2012\/23688-5","2008\/57528-9","2005\/56253-8","309649\/2021-8","303242\/2018-3","421254\/2016-5","311105\/2015-7","301318\/2019-0","303707\/2015-1","162944\/2020-9","001"],"award-info":[{"award-number":["2020\/02415-7","2018\/14685-9","2021\/09519-5","2019\/14038-6","2017\/14414-2","2012\/23688-5","2008\/57528-9","2005\/56253-8","309649\/2021-8","303242\/2018-3","421254\/2016-5","311105\/2015-7","301318\/2019-0","303707\/2015-1","162944\/2020-9","001"]}]},{"name":"National Council for Scientific and Technological Development","award":["2020\/02415-7","2018\/14685-9","2021\/09519-5","2019\/14038-6","2017\/14414-2","2012\/23688-5","2008\/57528-9","2005\/56253-8","309649\/2021-8","303242\/2018-3","421254\/2016-5","311105\/2015-7","301318\/2019-0","303707\/2015-1","162944\/2020-9","001"],"award-info":[{"award-number":["2020\/02415-7","2018\/14685-9","2021\/09519-5","2019\/14038-6","2017\/14414-2","2012\/23688-5","2008\/57528-9","2005\/56253-8","309649\/2021-8","303242\/2018-3","421254\/2016-5","311105\/2015-7","301318\/2019-0","303707\/2015-1","162944\/2020-9","001"]}]},{"DOI":"10.13039\/501100002322","name":"Coordination for the Improvement of Higher Level Personnel","doi-asserted-by":"publisher","award":["2020\/02415-7","2018\/14685-9","2021\/09519-5","2019\/14038-6","2017\/14414-2","2012\/23688-5","2008\/57528-9","2005\/56253-8","309649\/2021-8","303242\/2018-3","421254\/2016-5","311105\/2015-7","301318\/2019-0","303707\/2015-1","162944\/2020-9","001"],"award-info":[{"award-number":["2020\/02415-7","2018\/14685-9","2021\/09519-5","2019\/14038-6","2017\/14414-2","2012\/23688-5","2008\/57528-9","2005\/56253-8","309649\/2021-8","303242\/2018-3","421254\/2016-5","311105\/2015-7","301318\/2019-0","303707\/2015-1","162944\/2020-9","001"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the dynamics of classical particles confined in a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear discrete mapping for the variables energy en and phase \u03d5n of the periodic moving well. We obtain the phase space and show that it contains periodic islands, chaotic sea, and invariant spanning curves. We find the elliptic and hyperbolic fixed points and discuss a numerical method to obtain them. We study the dispersion of the initial conditions after a single iteration. This study allows finding regions where multiple reflections occur. Multiple reflections happen when a particle does not have enough energy to exit the potential well and is trapped inside it, suffering several reflections until it has enough energy to exit. We also show deformations in regions with multiple reflection, but the area remains constant when we change the control parameter NC. Finally, we show some structures that appear in the e0e1 plane by using density plots.<\/jats:p>","DOI":"10.3390\/e24101427","type":"journal-article","created":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T04:04:56Z","timestamp":1665201896000},"page":"1427","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multiple Reflections for Classical Particles Moving under the Influence of a Time-Dependent Potential Well"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3906-8267","authenticated-orcid":false,"given":"Fl\u00e1vio Heleno","family":"Graciano","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual Paulista (UNESP), Instituto de Geoci\u00eancias e Ci\u00eancias Exatas, C\u00e2mpus de Rio Claro, Av. 24A, 1515, S\u00e3o Paulo 13506-900, SP, Brazil"},{"name":"Instituto Federal do Sul de Minas Gerais (IFSULDEMINAS), Campus Pouso Alegre, Avenida Maria da Concei\u00e7\u00e3o Santos n\u00ba 900, Bairro Parque Real, Pouso Alegre 37560-260, MG, Brazil"}]},{"given":"Diogo Ricardo","family":"da Costa","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual Paulista (UNESP), Instituto de Geoci\u00eancias e Ci\u00eancias Exatas, C\u00e2mpus de Rio Claro, Av. 24A, 1515, S\u00e3o Paulo 13506-900, SP, Brazil"},{"name":"Departamento de F\u00edsica, Universidade Federal do Paran\u00e1 (UFPR), Curitiba 80060-000, PR, Brazil"},{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica da Universidade de S\u00e3o Paulo (IME-USP), Rua do Mat\u00e3o, 1010, S\u00e3o Paulo 05508-090, SP, Brazil"}]},{"given":"Edson D.","family":"Leonel","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual Paulista (UNESP), Instituto de Geoci\u00eancias e Ci\u00eancias Exatas, C\u00e2mpus de Rio Claro, Av. 24A, 1515, S\u00e3o Paulo 13506-900, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7238-4777","authenticated-orcid":false,"given":"Juliano A.","family":"de Oliveira","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual Paulista (UNESP), Instituto de Geoci\u00eancias e Ci\u00eancias Exatas, C\u00e2mpus de Rio Claro, Av. 24A, 1515, S\u00e3o Paulo 13506-900, SP, Brazil"},{"name":"C\u00e2mpus de S\u00e3o Jo\u00e3o da Boa Vista, Universidade Estadual Paulista, Av. Profa. Isette Corr\u00eaa Font\u00e3o, 505, S\u00e3o Paulo 13876-750, SP, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,7]]},"reference":[{"key":"ref_1","unstructured":"Devaney, R.L. (1991). A First Course in Chaotic Dynamical System, CRC Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1103\/PhysRev.75.1169","article-title":"On the Origin of the Cosmic Radiation","volume":"75","author":"Fermi","year":"1949","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"026201","DOI":"10.1103\/PhysRevE.73.026201","article-title":"Time-dependent properties of a simplified Fermi-Ulam accelerator model","volume":"73","author":"Ladeira","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ulam, S.M. (1961). 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