{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:26:37Z","timestamp":1787232397326,"version":"build-2736575974"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>We present an algorithm to compute the one-dimensional stable manifold of a saddle point for a planar map. In contrast to current standard techniques, here it is not necessary to know the inverse or approximate it, for example, by using Newton's method. Rather than using the inverse, the manifold is grown starting from the linear eigenspace near the saddle point by adding a point that maps back onto an earlier segment of the stable manifold. The performance of the algorithm is compared to other methods using an example in which the inverse map is known explicitly. The strength of our method is illustrated with examples of noninvertible maps, where the stable set may consist of many different pieces, and with a piecewise-smooth model of an interrupted cutting process. The algorithm has been implemented for use in the DsTool environment and is available for download with this paper.<\/jats:p>","DOI":"10.1137\/030600131","type":"journal-article","created":{"date-parts":[[2004,6,2]],"date-time":"2004-06-02T21:00:53Z","timestamp":1086210053000},"page":"161-190","source":"Crossref","is-referenced-by-count":67,"title":["Computing One-Dimensional Stable Manifolds and Stable Sets of Planar Maps without the Inverse"],"prefix":"10.1137","volume":"3","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":"10.1007\/978-1-4612-1936-1"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58124-3"},{"key":"R3","first-page":"303","volume":"39","author":"Back A.","year":"1992","journal-title":"Notices Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9920","issn-type":"print"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0007-8506(07)62891-1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1115\/1.1455030"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050240"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497000972"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(02)00751-0"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"R10","unstructured":"I. Gumowski, C. Mira, Dynamique chaotique, Cepadues \u00c9ditions, 1980, 480\u20130, Transformations ponctuelles. Transition. Ordre\u2013d\u00e9sordre. [Point transformations. Transition. Order\u2013disorder]; Collection NABLA83j:58002"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089135"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAB.2.000552"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1993.1002"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF00944751"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/0030-4018(79)90090-7"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(95)00309-6"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6059"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000310"},{"key":"R19","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":"R20","unstructured":"Maplesoft Product Suite, http:\/\/www.maplesoft.com\/ (2004)."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1142\/S021812749600148X"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1142\/9789812798732"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(98)00234-6"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000279"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(89)90253-4"},{"key":"R26","unstructured":"H. M. Osinga and J. P. England,\n                      Global Manifold 1D Code, Version\n                      2, Software for Use with DsTool, http:\/\/www.dynamicalsystems.org\/sw\/sw\/detail?item=27 (2003)."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5703-5"},{"key":"R28","volume-title":"Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations","author":"Palis Jacob","year":"1993"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3486-9"},{"key":"R30","unstructured":"C. Sim\u00f3,\n                      On the analytical and numerical approximation of invariant manifolds\n                      , in Les M\u00e9thodes Modernes de la M\u00e9canique C\u00e9leste, D. Benest and C. Froeschl\u00e9, eds., Goutelas, 1989, pp. 285\u2013330."},{"key":"R31","unstructured":"S. H. Strogatz,\n                      Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering\n                      , Perseus Books, Reading, MA, Cambridge, MA, 1994."},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.86.2261"},{"key":"R33","unstructured":"R. Szalai,\n                      Nonlinear Vibrations of Interrupted Cutting Processes\n                      , M.Sc. thesis, Budapest University of Technology and Economics, Budapest, Hungary, 2002."},{"key":"R34","unstructured":"F. J. Wicklin,\n                      Pisces: A Platform for Implicit Surfaces and Curves and the Exploration of Singularities\n                      , Tech. report GCG 89, The Geometry Center, University of Minnesota, Minneapolis, MN, 1995. Available online at http:\/\/www.geom.uiuc.edu\/\u02dcfjw\/pisces\/."},{"key":"R35","doi-asserted-by":"crossref","unstructured":"Z. You, E. J. Kostelich, and J. A. Yorke,\n                      Calculating stable and unstable manifolds\n                      , 1 (1991), pp. 605\u2013623.","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\/030600131","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:46:53Z","timestamp":1787230013000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/030600131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1137\/030600131"],"URL":"https:\/\/doi.org\/10.1137\/030600131","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1]]}}}