{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:35:46Z","timestamp":1787330146186,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ODEs, namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark--Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic BVP that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS.<\/jats:p>","DOI":"10.1137\/s0036142902400779","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"401-435","source":"Crossref","is-referenced-by-count":43,"title":["Computation of Periodic Solution Bifurcations in ODEs Using Bordered Systems"],"prefix":"10.1137","volume":"41","author":[{"given":"E. J.","family":"Doedel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W.","family":"Govaerts","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu. A.","family":"Kuznetsov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","unstructured":"Eugene Allgower, Kurt Georg, Numerical path following, Handb. Numer. Anal., V, North\u2010Holland, Amsterdam, 1997, 3\u20132071470225"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1979-0521281-7"},{"key":"R3","unstructured":"Wolf\u2010J\u00fcrgen Beyn, Alan Champneys, Eusebius Doedel, Willy Govaerts, Yuri Kuznetsov, Bj\u00f6rn Sandstede, Numerical continuation, and computation of normal forms, North\u2010Holland, Amsterdam, 2002, 149\u20132191900655"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0710052"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(80)90090-6"},{"key":"R6","unstructured":"E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede, and X. J. Wang, auto97:\n                      Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)\n                      , Concordia University, Montreal, QC, Canada, 1997."},{"key":"R6","unstructured":"Available via ftp from ftp.cs.concordia.ca\/pub\/doedel\/auto."},{"key":"R7","unstructured":"E. Doedel, A. Jepson, H. Keller, Numerical methods for Hopf bifurcation and continuation of periodic solution paths, North\u2010Holland, Amsterdam, 1984, 127\u2013138806775"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000397"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000555"},{"key":"R10","unstructured":"T. F. Fairgrieve,\n                      The Computations and Use of Floquet Multipliers for Bifurcation Analysis\n                      , Ph.D. Thesis, University of Toronto, Toronto, ON, Canada, 1994."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1145\/293686.293693"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719543"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0721012"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"R15","unstructured":"A. D. Jepson,\n                      Numerical Hopf Bifurcation\n                      , Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1981."},{"key":"R16","unstructured":"H. B. Keller,\n                      Numerical solution of bifurcation and nonlinear eigenvalue problems\n                      , in Applications of Bifurcation Theory, P. H. Rabinowitz, ed., Academic Press, New York, 1977, pp. 359\u2013384."},{"key":"R17","volume-title":"Elements of applied bifurcation theory","author":"Kuznetsov Yuri","year":"1998"},{"key":"R18","unstructured":"Yu. A. Kuznetsov and V. V. Levitin, content:\n                      A Multiplatform Environment for Analyzing Dynamical Systems\n                      , Dynamical Systems Laboratory, CWI, Amsterdam, 1995\u20131997."},{"key":"R18","unstructured":"Available via ftp from ftp.cwi.nl\/pub\/content."},{"key":"R19","unstructured":"Yu. A. Kuznetsov, W. Mestrom, and A. M. Riet,\n                      A Continuation Toolbox in matlab\n                      , Mathematical Institute, Utrecht University, Utrecht, The Netherlands, 2001, http:\/\/www.math.uu.nl\/people\/kuznet\/cm."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/0717048"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0715004"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01398649"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0036142902400779","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:49:19Z","timestamp":1787327359000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0036142902400779"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0036142902400779"],"URL":"https:\/\/doi.org\/10.1137\/s0036142902400779","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}