{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:38:14Z","timestamp":1787236694094,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["PID-2021-122954NB-100"],"award-info":[{"award-number":["PID-2021-122954NB-100"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003741","name":"Instituci\u00f3 Catalana de Recerca i Estudis Avan\u00e7ats","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003741","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004895","name":"European Social Fund","doi-asserted-by":"publisher","award":["PRE2019-088132"],"award-info":[{"award-number":["PRE2019-088132"]}],"id":[{"id":"10.13039\/501100004895","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["CEX2020-001084-M"],"award-info":[{"award-number":["CEX2020-001084-M"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this work we study the splitting distance of a rapidly perturbed pendulum [Formula: see text] with [Formula: see text] a [Formula: see text]-periodic function and [Formula: see text]. Systems of this kind undergo exponentially small splitting, and, when [Formula: see text], it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided [Formula: see text]. Our study focuses on the case [Formula: see text], and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as [Formula: see text]. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency [Formula: see text] in Arnold\u2019s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with [Formula: see text] where, for most [Formula: see text], the perturbation satisfies [Formula: see text]. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton\u2013Jacobi formalism. The leading exponentially small term appears at order [Formula: see text], where [Formula: see text] is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.<\/jats:p>","DOI":"10.1137\/23m1550992","type":"journal-article","created":{"date-parts":[[2024,5,9]],"date-time":"2024-05-09T04:01:34Z","timestamp":1715227294000},"page":"1159-1198","source":"Crossref","is-referenced-by-count":1,"title":["Splitting of Separatrices for Rapid Degenerate Perturbations of the Classical Pendulum"],"prefix":"10.1137","volume":"23","author":[{"given":"Inmaculada","family":"Baldom\u00e1","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya, Av. Diagonal, 647 08028, Barcelona, Spain."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tere","family":"M-Seara","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya, Av. Diagonal, 647 08028, Barcelona, Spain."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7769-4942","authenticated-orcid":true,"given":"Rom\u00e1n","family":"Moreno","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya, Av. Diagonal, 647 08028, Barcelona, Spain."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,5,9]]},"reference":[{"key":"ref1","series-title":"London Math. Soc. Lecture Note Ser. 192","first-page":"5","volume-title":"Symplectic Geometry","author":"Angenent S.","year":"1993"},{"key":"ref2","first-page":"9","volume":"156","author":"Arnold V. I.","year":"1964","journal-title":"Dokl. Akad. 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Stud. 77","volume-title":"Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics","author":"Moser J.","year":"1973"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2021.12.015"},{"key":"ref19","unstructured":"C. Oliv\u00e9 Farr\u00e9, C\u00e0lcul de l\u2019escissi\u00f3 de separatrius usant t\u00e8cniques de matching complex i ressurg\u00e8ncia aplicades a l\u2019equaci\u00f3 de Hamilton-Jacobi, Ph.D. thesis, Facultat de Matem\u00e0tiques i Estad\u00edstica, Universitat Polit\u00e8cnica de Catalunya, 2006, http:\/\/hdl.handle.net\/2117\/93163."},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-9593(00)01063-6"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/097\/1021048"},{"key":"ref22","unstructured":"Q. Wang, Non-dominance of the Melnikov Function: An Example, preprint, University of Arizona, 2020."}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1550992","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:53:40Z","timestamp":1787234020000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1550992"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,9]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,6,30]]}},"alternative-id":["10.1137\/23M1550992"],"URL":"https:\/\/doi.org\/10.1137\/23m1550992","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,9]]}}}