{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:27:33Z","timestamp":1787232453928,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2005,1]]},"abstract":"<jats:p>We present an algorithm for computing one-dimensional stable and unstable manifolds of saddle periodic orbits in a Poincar\u00e9 section. The computation is set up as a boundary value problem by restricting both end points of orbit segments to the section. Starting from the periodic orbit itself, we use collocation routines from {\\sc Auto} to continue the solutions of the boundary value problem such that one end point of the orbit segment varies along a part of the manifold that was already computed. In this way, the other end point of the orbit segment traces out a new piece of the manifold.<\/jats:p>\n                  <jats:p>As opposed to standard methods that use shooting to compute the Poincar\u00e9 map as the kth return map, our approach defines the Poincar\u00e9 map as the solution of a boundary value problem. This enables us to compute global manifolds through points where the flow is tangent to the section---a situation that is typically encountered unless one is dealing with a periodically forced system. Another major advantage of our approach is that it deals effectively with the problem of extreme sensitivity of the Poincar\u00e9 map to its argument, which is a typical feature in the important class of slow-fast systems.<\/jats:p>\n                  <jats:p>We illustrate and test our algorithm by computing stable and unstable manifolds for three examples: the forced Van der Pol oscillator, a model of a semiconductor laser with optical injection, and a slow-fast chemical oscillator. All examples are accompanied by animations demonstrating how the manifolds grow during the computation.<\/jats:p>","DOI":"10.1137\/05062408x","type":"journal-article","created":{"date-parts":[[2005,11,3]],"date-time":"2005-11-03T21:00:19Z","timestamp":1131051619000},"page":"1008-1041","source":"Crossref","is-referenced-by-count":34,"title":["Computing One-Dimensional Global Manifolds of Poincar\u00e9 Maps by Continuation"],"prefix":"10.1137","volume":"4","author":[{"given":"J. P.","family":"England","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B.","family":"Krauskopf","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. M.","family":"Osinga","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,7]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1021\/j100442a009"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S1111111102419130"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.056201"},{"key":"R4","unstructured":"Eusebius Doedel, AUTO: a program for the automatic bifurcation analysis of autonomous systems, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), Vol. 30, 1981, 265\u201328484b:58001"},{"key":"R5","unstructured":"E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. E. Oldeman, B. Sandstede, and X. J. Wang,\n                      AUTO2000: Continuation and Bifurcation Software for Ordinary Differential Equations\n                      , available online at http:\/\/cmvl.cs.concordia.ca\/."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/030600131"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405012466"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1038\/scientificamerican0383-112"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.11.018"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"R11","volume-title":"Differential equations, dynamical systems, and linear algebra","author":"Hirsch Morris","year":"1974"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1993.1002"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(94)00171-L"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000310"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6059"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"Bernd Krauskopf, Hinke Osinga, Investigating torus bifurcations in the forced van der Pol oscillator, IMA Vol. Math. Appl., Vol. 119, Springer, New York, 2000, 199\u20132082001b:37118","DOI":"10.1007\/978-1-4612-1208-9_9"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1063\/1.166450"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/030600180"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405012533"},{"key":"R20","unstructured":"Yu. A. Kuznetsov,\n                      CONTENT\u2014Integrated Environment for Analysis of Dynamical Systems. Tutorial\n                      , \u00c9cole Normale Sup\u00e9rieure de Lyon, Rapport de Recherche UPMA\u201098\u2010224, 1998."},{"key":"R21","unstructured":"H. M. Osinga and J. P. England,\n                      Separating manifolds in slow\u2010fast systems\n                      , in Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, 2005, D. van Campen, M. Lazurko, and W. van der Oever, eds., ID 15\u2010454 (CD), pp. 1699\u20131705."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5703-5"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3486-9"},{"key":"R24","first-page":"701","volume":"1","author":"Van der Pol B.","year":"1920","journal-title":"Radio Review"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"S. H. Strogatz,\n                      Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering\n                      , Addison\u2010Wesley, Reading, MA, 1994.","DOI":"10.1063\/1.4823332"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/S0030-4018(99)00603-3"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2005.06.003"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(99)00088-0"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000440"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/05062408X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:55:12Z","timestamp":1787230512000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/05062408X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2005,1]]}},"alternative-id":["10.1137\/05062408X"],"URL":"https:\/\/doi.org\/10.1137\/05062408x","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,1]]}}}