{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:32:33Z","timestamp":1787239953468,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100031890","name":"New Jersey Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100031890","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100007812","name":"University of Washington","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007812","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:p>Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. In this module, we formulate and prove the one-dimensional analogues of Peixoto's theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible one-dimensional treatment of structural stability theory that should help make Peixoto's theorems and their more recent generalizations easier to appreciate and understand. Further, we believe it is important and interesting for students to know the historical context of these discoveries since the mathematics was not done in isolation. The historical context is perhaps even more appropriate as it is the 100th anniversary of Mar\u00edlia Chaves Peixoto's and Maur\u00edcio Matos Peixoto's births, February 24th and April 15th, 1921, respectively.<\/jats:p>","DOI":"10.1137\/21m1426572","type":"journal-article","created":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T04:42:43Z","timestamp":1691469763000},"page":"869-886","source":"Crossref","is-referenced-by-count":2,"title":["The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof"],"prefix":"10.1137","volume":"65","author":[{"given":"Aminur","family":"Rahman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Blackmore","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,8,8]]},"reference":[{"key":"atypb1","first-page":"247","volume":"14","author":"Andronov A.","year":"1937","journal-title":"Dokl. Akad. Nauk. SSSR"},{"key":"atypb2","volume-title":"annotationes et additiones ad ea, quae in actis sup. de curva elastica, iisochrona paracentrica, et velaria, hinc inde memorata, et paratim controversa legundur","author":"Bernoulli J."},{"key":"atypb3","volume-title":"Differential Equations","author":"Blanchard P.","year":"2011","edition":"4"},{"key":"atypb4","volume-title":"Differential Topology","author":"Guillemin V.","year":"1974"},{"key":"atypb5","unstructured":"G. W. Leibniz,\n                      Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus\n                      , Acta Eruditorum, 1684."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718232"},{"key":"atypb7","volume-title":"Topology from the Differentiable Viewpoint","author":"Milnor J.","year":"1997"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"I. Newton,\n                      Philosophi\\ae Naturalis Principia Mathematica\n                      , The Royal Society, 1687.","DOI":"10.5479\/sil.52126.39088015628399"},{"key":"atypb9","unstructured":"I. Newton,\n                      Methodus fluxionum et serierum infinitarum cum eisudem applicatione ad curvarum geometriam\n                      , Opuscola Mathematica Philosophica et Philologica, 1744."},{"key":"atypb10","first-page":"135","volume":"31","author":"Peixoto M. C.","year":"1959","journal-title":"An. Acad. Brasil. Ci."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.2307\/1970100"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(65)90018-2"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0003-8"},{"key":"atypb14","volume-title":"Les m\u00e9thodes nouvelles de la m\u00e9canique c\u00e9leste","volume":"1","author":"Poincar\u00e9 J. 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Math."},{"key":"atypb18","volume-title":"Nonlinear Dynamics and Chaos","author":"Strogatz S.","year":"1994"}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1426572","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:10:47Z","timestamp":1787238647000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1426572"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["10.1137\/21M1426572"],"URL":"https:\/\/doi.org\/10.1137\/21m1426572","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8]]}}}