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For following bifurcation curves in two parameters, the pseudo\u2010arclength method is used combined with Newton iteration. Two test examples (an interrupted machining model and a traffic model with driver reaction time) conclude the paper. The algorithm has been implemented in the software tool PDDE\u2010cont.<\/jats:p>","DOI":"10.1137\/040618709","type":"journal-article","created":{"date-parts":[[2006,8,24]],"date-time":"2006-08-24T15:30:58Z","timestamp":1156433458000},"page":"1301-1317","source":"Crossref","is-referenced-by-count":55,"title":["Continuation of Bifurcations in Periodic Delay\u2010Differential Equations Using Characteristic Matrices"],"prefix":"10.1137","volume":"28","author":[{"given":"R\u00f3bert","family":"Szalai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G\u00e1bor","family":"St\u00e9p\u00e1n","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"John Hogan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","unstructured":"W. J. Beyn, A. Champneys, E. J. Doedel, W. Govaerts, B. Sandstede, and Y. A. Kuznetsov,\n                      Numerical continuation and computation of normal forms\n                      , in Handbook of Dynamical Systems, B. Fiedler, ed., North\u2013Holland, Amsterdam, 2002, pp. 149\u2013219."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457866"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500026731"},{"key":"R4","unstructured":"T. A. Davies,\n                      UMFPACK Version 4.1 User Guide\n                      , Technical report, Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 2003;"},{"key":"R4","unstructured":"available online from http:\/\/www.cise.ufl.edu\/research\/sparse\/umfpack\/."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0710052"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1145\/779359.779362"},{"key":"R7","unstructured":"O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther,\n                      Delay Equations\n                      , Springer\u2010Verlag, New York, 1995."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902400779"},{"key":"R9","unstructured":"E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X.\u2010J. 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Hale,\n                      Theory of Functional Differential Equations\n                      , Springer\u2010Verlag, New York, 1977."},{"key":"R18","unstructured":"J. K. Hale and S. M. Verduyn Lunel,\n                      Introduction to Functional Differential Equations\n                      , Springer\u2010Verlag, New York, 1993."},{"key":"R19","unstructured":"E. Hille and R. S. Phillips,\n                      Functional Analysis and Semigroups\n                      , AMS, Providence, RI, 1957."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1992-1155350-0"},{"key":"R21","unstructured":"Y. A. Kuznetsov,\n                      CONTENT\u2014Integrated Environment for Analysis of Dynamical Systems: Tutorial\n                      , Rapport de Recherche UPMA\u201098\u2010224, Ecole Normale Superieure de Lyon, Lyon, France, 1998."},{"key":"R22","unstructured":"S. M. Verduyn Lunel,\n                      Small solutions and completeness for linear functional differential equations\n                      , in Oscillations and Dynamics in Delay Equations, Contemp. Math. 129, AMS, Providence, RI, 1992, pp. 127\u2013152."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.70.026207"},{"key":"R24","unstructured":"R. Szalai,\n                      PDDE\u2010CONT: A Continuation and Bifurcation Software for Delay\u2010Differential Equations\n                      , Technical report, Budapest University of Technology and Economics, Budapest, Hungary, 2005;"},{"key":"R24","unstructured":"available online from http:\/\/www.mm.bme.hu\/\u02dcszalai\/pdde\/."},{"key":"R25","unstructured":"R. Szalai and G. St\u00e9p\u00e1n,\n                      Stability boundaries of high\u2010speed milling corresponding to period doubling are essentially closed curves\n                      , in Proceedings of IMECE2003, ASME International Mechanical Engineering Congress and R&D Expo, 2003, pp. 1\u20136 (CD\u2010ROM)."},{"key":"R26","unstructured":"R. Szalai and G. St\u00e9p\u00e1n,\n                      Period Doubling Bifurcation and Center Manifold Reduction in a Time\u2010Periodic and Time\u2010Delayed Model of Machining\n                      , preprint, 2006;"},{"key":"R26","unstructured":"available online from http:\/\/www.mm.bme.hu\/\u02dcszalai\/publications.html."},{"key":"R27","unstructured":"J. Tlusty,\n                      Manufacturing Process and Equipment\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 2000."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/040618709","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:54:48Z","timestamp":1787334888000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/040618709"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/040618709"],"URL":"https:\/\/doi.org\/10.1137\/040618709","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}