{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:41:35Z","timestamp":1776724895124,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"227","license":[{"start":{"date-parts":[[2000,2,13]],"date-time":"2000-02-13T00:00:00Z","timestamp":950400000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -D restrictions of the bifurcation equation. The test functions that characterise each of the equivalence classes are constructed by means of an equivariant numerical version of the Liapunov-Schmidt reduction. The classification supplies limited qualitative information concerning the imperfect bifurcation diagrams of the detected bifurcation points.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01052-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1097-1108","source":"Crossref","is-referenced-by-count":5,"title":["Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies"],"prefix":"10.1090","volume":"68","author":[{"given":"Klaus","family":"B\u00f6hmer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Willy","family":"Govaerts","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladim\u00edr","family":"Janovsk\u00fd","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,2,13]]},"reference":[{"key":"1","unstructured":"K. B\u00f6hmer, W. Govaerts and V. Janovsk\u00fd, Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies, Bericht zur Fachbereich Mathematik der Philipps-Universit\u00e4t Marburg 1996."},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF02238535","article-title":"On a numerical Liapunov-Schmidt method for operator equations","volume":"51","author":"B\u00f6hmer, K.","year":"1993","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"1-2","key":"3","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/0377-0427(89)90150-7","article-title":"Computational methods for bifurcation problems with symmetries\u2014with special attention to steady state and Hopf bifurcation points","volume":"26","author":"Dellnitz, Michael","year":"1989","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"4","unstructured":"K. Gatermann and R. Lauterbach, Automatic classification of normal forms, Preprint SC 95-3 (Februar 1995), Konrad-Zuse-Zentrum f\u00fcr Informationstechnik Berlin, Germany"},{"key":"5","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5034-0","volume-title":"Singularities and groups in bifurcation theory. Vol. I","volume":"51","author":"Golubitsky, Martin","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0387909990"},{"key":"6","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4574-2","volume-title":"Singularities and groups in bifurcation theory. Vol. II","volume":"69","author":"Golubitsky, Martin","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0387966528"},{"key":"7","doi-asserted-by":"crossref","unstructured":"W. Govaerts, Computation of singularities in large nonlinear systems, SIAM J. Num. Anal. 34 (1997) pp.  867\u2013880.","DOI":"10.1137\/S0036142994272167"},{"key":"8","doi-asserted-by":"crossref","unstructured":"V. Janovsk\u00fd and P. Plech\u00e1\u010d, Numerical applications of equivariant reduction techniques, in Bifurcation and Symmetry, M. Golubitsky. E. Allgower, K. B\u00f6hmer, ed., Birkh\u00e4ser Verlag, 1992.","DOI":"10.1007\/978-3-0348-7536-3_18"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1137\/0520004","article-title":"On a reduction process for nonlinear equations","volume":"20","author":"Jepson, A. D.","year":"1989","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"10","isbn-type":"print","first-page":"443","article-title":"The numerical analysis of bifurcation problems with symmetries based on bordered Jacobians","author":"Werner, Bodo","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0821811347"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01052-2\/S0025-5718-99-01052-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01052-2\/S0025-5718-99-01052-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:05:49Z","timestamp":1776722749000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01052-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,2,13]]},"references-count":10,"journal-issue":{"issue":"227","published-print":{"date-parts":[[1999,7]]}},"alternative-id":["S0025-5718-99-01052-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01052-2","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,2,13]]}}}