{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:34Z","timestamp":1787337214944,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a MATLAB toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., limit point, period-doubling, and Neimark\u2013Sacker) and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized H\u00e9non map and a juvenile\/adult competition model from mathematical biology.<\/jats:p>","DOI":"10.1137\/060653858","type":"journal-article","created":{"date-parts":[[2007,11,2]],"date-time":"2007-11-02T18:30:30Z","timestamp":1194028230000},"page":"2644-2667","source":"Crossref","is-referenced-by-count":85,"title":["Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps"],"prefix":"10.1137","volume":"29","author":[{"given":"W.","family":"Govaerts","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R. Khoshsiar","family":"Ghaziani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu. A.","family":"Kuznetsov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. G. E.","family":"Meijer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,11,2]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"E. L. Allgower and K. Georg,\n                      Numerical Continuation Methods: An Introduction\n                      , Springer-Verlag, Berlin, 1990.","DOI":"10.1007\/978-3-642-61257-2"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"V. I. Arnol'd,\n                      Geometrical Methods in the Theory of Ordinary Differential Equations\n                      , Springer-Verlag, New York, 1983.","DOI":"10.1007\/978-1-4684-0147-9"},{"key":"R3","unstructured":"A. Back, J. Guckenheimer, M. Myers, F. Wicklin, and P. Worfolk,\n                      DsTool: Computer assisted exploration of dynamical systems\n                      , Notices Amer. Math. Soc., 39 (1992),pp. 303\u2013309."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142995281693"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"W.J. Beyn, A. Champneys, E. Doedel, W. Govearts, Yu. A. Kuznetsov, and B. Sandstede,\n                      Numerical continuation, and computation of normal forms\n                      , in Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed., Elsevier Science, Amsterdam, 2002,pp. 149\u2013219.","DOI":"10.1016\/S1874-575X(02)80025-X"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000117"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"A. Dhooge, W. Govaerts, Yu. A. Kuznetsov, W. Mestrom, and A. M. Riet,\n                      CL_MATCONT: A continuation toolbox in MATLAB\n                      , in Proceedings of the 2003 ACM Symposium on Applied Computing, Melbourne, FL, 2003, Association for Computing Machinery, New York, pp. 161\u2013166.","DOI":"10.1145\/952532.952567"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1145\/779359.779362"},{"key":"R9","unstructured":"E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede, and X.J. Wang, AUTO97-AUTO2000:\n                      Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), User's Guide\n                      , Concordia University, Montreal, Canada (1997\u20132000), available online from http:\/\/indy.cs.concordia.ca."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1046\/j.1365-2656.2003.00743.x"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/04060487X"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"W. Govaerts,Yu. A. Kuznetsov, and B. Sijnave,\n                      Bifurcations of maps in the software package\n                      CONTENT, in Computer Algebra in Scientific Computing\u2014CASC'99 (Munich), V. G. Ganzha, V. G. E. W. Mayr, and E. V. Vorozhtsov, eds., Springer, Berlin, 1999, pp. 191\u2013206.","DOI":"10.1007\/978-3-642-60218-4_14"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"W. J. F. Govaerts,\n                      Numerical Methods for Bifurcations of Dynamical Equilibria\n                      , SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719543"},{"key":"R14","unstructured":"B. D. Hassard, N. D. Kazarinoff, and Y.H. Wan,\n                      Theory and Applications of Hopf Bifurcation\n                      , Cambridge University Press, Cambridge, UK, 1981."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(93)90294-B"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6059"},{"key":"R17","unstructured":"Yu. A. Kuznetsov and V. V. Levitin, CONTENT:\n                      A Multiplatform Environment for Analyzing Dynamical Systems\n                      , CWI, Amsterdam, 1997; available online via ftp from ftp.cwi.nl\/pub\/CONTENT."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"Yu. A. Kuznetsov,\n                      Elements of Applied Bifurcation Theory\n                      , 3rd ed., Springer-Verlag, New York, 2004.","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/030601508"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1080\/10236190600986842"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/45.3-4.316"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.2307\/2708"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"C. Mira,\n                      Chaotic Dynamics\n                      , World Scientific, Singapore, 1987.","DOI":"10.1142\/0413"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"H. Nusse and J. Yorke,\n                      Dynamics: Numerical Explorations\n                      , 2nd ed., Springer-Verlag, New York, 1998.","DOI":"10.1007\/978-1-4612-0219-6"},{"key":"R25","unstructured":"Y. L. Song,\n                      The Juvenile\/Adult Leslie\u2013Gower Competition Model\n                      , preprint, 2004."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/0145053"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060653858","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:49:56Z","timestamp":1787334596000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060653858"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":26,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/060653858"],"URL":"https:\/\/doi.org\/10.1137\/060653858","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}