{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:35:37Z","timestamp":1776800137750,"version":"3.51.2"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"297","license":[{"start":{"date-parts":[[2016,6,2]],"date-time":"2016-06-02T00:00:00Z","timestamp":1464825600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Many applications lead to separable parameterized equations of the form\n \n \n \n \n F<\/mml:mi>\n (<\/mml:mo>\n y<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n ,<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n \n \u2261\n \n <\/mml:mo>\n A<\/mml:mi>\n (<\/mml:mo>\n y<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n z<\/mml:mi>\n +<\/mml:mo>\n b<\/mml:mi>\n (<\/mml:mo>\n y<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n F(y,\\mu ,z) \\equiv A(y, \\mu )z+b(y, \\mu )=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n y<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n y \\in \\mathbb R^n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n z<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n N<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n z \\in \\mathbb R^N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n A<\/mml:mi>\n (<\/mml:mo>\n y<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n (<\/mml:mo>\n N<\/mml:mi>\n +<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n \n \u00d7\n \n <\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n A(y, \\mu ) \\in \\mathbb {R}^{(N+n) \\times N}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n b<\/mml:mi>\n (<\/mml:mo>\n y<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n N<\/mml:mi>\n +<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n b(y, \\mu ) \\in \\mathbb {R}^{N+n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \u03bc\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \\mu \\in \\mathbb R<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a parameter. Typically\n \n \n \n \n N<\/mml:mi>\n >><\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n N >>n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Suppose bifurcation occurs at a solution point\n \n \n \n \n (<\/mml:mo>\n \n y<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \n \u03bc\n \n <\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n ,<\/mml:mo>\n \n z<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n (y^*,\\mu ^*,z^*)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of this equation. If\n \n \n \n \n A<\/mml:mi>\n (<\/mml:mo>\n \n y<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \n \u03bc\n \n <\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n A(y^*, \\mu ^*)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is rank deficient, then the linear component\n \n \n \n z<\/mml:mi>\n z<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n bifurcates into an affine subspace at this point. We show how to compute such a point\n \n \n \n \n (<\/mml:mo>\n y<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n ,<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (y,\\mu ,z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by reducing the original system to a smaller separable system, while preserving the bifurcation, the rank deficiencies and a non-degeneracy condition. A numerical algorithm for solving the reduced system and examples illustrating the method are provided.\n <\/p>","DOI":"10.1090\/mcom\/2968","type":"journal-article","created":{"date-parts":[[2015,6,2]],"date-time":"2015-06-02T14:11:53Z","timestamp":1433254313000},"page":"271-293","source":"Crossref","is-referenced-by-count":3,"title":["Rank deficiencies and bifurcation into affine subspaces for separable parameterized equations"],"prefix":"10.1090","volume":"85","author":[{"given":"Yun-Qiu","family":"Shen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tjalling","family":"Ypma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,6,2]]},"reference":[{"key":"1","isbn-type":"print","first-page":"175","article-title":"Numerical methods for dynamical systems","author":"Beyn, Wolf-J\u00fcrgen","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0198534388"},{"key":"2","series-title":"Grundlehren der Mathematischen Wissenschaften","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-8159-4","volume-title":"Methods of bifurcation theory","volume":"251","author":"Chow, Shui Nee","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906649"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/0096-3003(88)90139-7","article-title":"Bifurcations via singular value decompositions","volume":"28","author":"Chow, Shui-Nee","year":"1988","journal-title":"Appl. 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