{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:26:19Z","timestamp":1787232379671,"version":"build-2736575974"},"reference-count":39,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. This paper introduces a rigorous numerical method to compute enclosures of the slow manifold of a slow-fast system with one fast and two slow variables. A triangulated first order approximation to the two dimensional invariant manifold is computed \u201calgebraically.\u201d Two translations of the computed manifold in the fast direction that are transverse to the vector field are computed as the boundaries of an initial enclosure. The enclosures are refined to bring them closer to each other by moving vertices of the enclosure boundaries one at a time. As an application we use the method to prove the existence of tangencies of invariant manifolds in the problem of singular Hopf bifurcation and to give bounds on the location of one such tangency.<\/jats:p>","DOI":"10.1137\/120861813","type":"journal-article","created":{"date-parts":[[2012,8,8]],"date-time":"2012-08-08T10:15:11Z","timestamp":1344420911000},"page":"831-863","source":"Crossref","is-referenced-by-count":5,"title":["Rigorous Enclosures of a Slow Manifold"],"prefix":"10.1137","volume":"11","author":[{"given":"John","family":"Guckenheimer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tomas","family":"Johnson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philipp","family":"Meerkamp","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,8,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/050626429"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.12.003"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2009.23.705"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2011.30.641"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/100791233"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1063\/1.2799471"},{"key":"atypb7","unstructured":"E. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang,\n                      AUTO: Software for continuation and bifurcation problems in ordinary differential equations\n                      , available at http:\/\/cmvl.cs.concordia.ca\/auto\/, 2010."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127407017562"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.1995.10504316"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/080718528"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/080741999"},{"key":"atypb13","unstructured":"J. Guckenheimer and P. Meerkamp,\n                      Unfoldings of Singular Hopf Bifurcation\n                      , preprint, http:\/\/arxiv. org\/abs\/1107.3185, 2011."},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"M. W. Hirsch, C. C. Pugh, and M. Shub,\n                      Invariant Manifolds\n                      , Lecture Notes in Math. 583, Springer-Verlag, Berlin, New York, 1977.","DOI":"10.1007\/BFb0092042"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"E. Hairer and G. Wanner,\n                      Solving Ordinary Differential Equations.II.Stiff and Differential-Algebraic Problems\n                      , 2nd ed., Springer Ser. Comput. Math., Springer-Verlag, Berlin, 1996.","DOI":"10.1007\/978-3-642-05221-7_1"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127400000840"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s12346-011-0040-2"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-011-0318-4"},{"key":"atypb19","unstructured":"T. Johnson and W. Tucker,\n                      On a fast and accurate method to enclose all zeros of an analytic function on a triangulated domain\n                      , in Proceedings of PARA 2008, Lecture Notes in Comput. Sci. 6126\/6127, Springer-Verlag, 2011, to appear."},{"key":"atypb20","first-page":"44","author":"C. K. R.","year":"1995","journal-title":"Berlin"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"A. Katok and B. Hasselblatt,\n                      Introduction to the Modern Theory of Dynamical Systems\n                      , Cambridge University Press, Cambridge, UK, 1995.","DOI":"10.1017\/CBO9780511809187"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(95)90061-6"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405012533"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/070688912"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1063\/1.3615231"},{"key":"atypb26","first-page":"175","author":"R","year":"2001","journal-title":"RI"},{"key":"atypb27","unstructured":"R. Lohner,\n                      Einschlie\u00dfung der Lo\u0308sung gewo\u0308hnlicher Anfangs- und Randwertaufgaben und Anwendungen\n                      , Ph.D. thesis, Universita\u0308t Karlsruhe, Karlsruhe, Germany, 1988."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-010-9083-z"},{"key":"atypb29","unstructured":"R. E. Moore,\n                      Interval Analysis\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1966."},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009936607335"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(98)10083-8"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1023\/A:1014798618404"},{"key":"atypb34","unstructured":"A. Neumaier,\n                      Interval Methods for Systems of Equations\n                      , Encyclopedia of Mathematics and Its Applications 37, Cambridge University Press, Cambridge, UK, 1990."},{"key":"atypb35","first-page":"129","volume":"32","author":"Ombach J.","year":"1995","journal-title":"Univ. Iagel. Acta Math."},{"key":"atypb37","first-page":"285","author":"Simo\u0301 C.","year":"1990","journal-title":"Paris"},{"key":"atypb38","doi-asserted-by":"crossref","unstructured":"W. Tucker,\n                      Validated Numerics. A Short Introduction to Rigorous Computations\n                      , Princeton University Press, Princeton, NJ, 2011.","DOI":"10.1515\/9781400838974"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/s102080010025"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.12.019"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.03.013"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1137\/090759975"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120861813","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:43:56Z","timestamp":1787229836000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120861813"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1137\/120861813"],"URL":"https:\/\/doi.org\/10.1137\/120861813","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1]]}}}