{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T10:23:42Z","timestamp":1775471022942,"version":"3.50.1"},"reference-count":42,"publisher":"World Scientific Pub Co Pte Lt","issue":"09","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2004,9]]},"abstract":"<jats:p> We consider stochastic delay differential equations of the form [Formula: see text] interpreted in the It\u00f4 sense, with Y(t)=\u03a6(t) for t\u2208[t<jats:sub>0<\/jats:sub>-\u03c4,t<jats:sub>0<\/jats:sub>] (here, W(t) is a standard Wiener process and \u03c4&gt;0 is the constant \"lag\", or \"time-lag\"). We are interested in bifurcations (that is, changes in the qualitative behavior of solutions of these equations) and we draw on insights from the related deterministic delay differential equation, for which there is a substantial body of known theory, and numerical results that enable us to discuss where changes occur in the behavior of the (exact and approximate) solutions of the equation. <\/jats:p><jats:p> Rather diverse components of mathematical background are necessary to understand the questions of interest. In this paper we first review some deterministic results and some basic elements of the stochastic analysis that (i) suggests lines of investigation for the stochastic case and (ii) are expected to facilitate the theoretical investigation of the stochastic problem. We then present the results of numerical experiments that illustrate some of the complexities that arise when considering bifurcations in stochastic delay differential equations. They give prima facie evidence for certain convergence properties of the bifurcation points estimated using the Euler\u2013Maruyama method for the equations considered. We conclude by drawing attention to a number of open questions in the field. <\/jats:p>","DOI":"10.1142\/s0218127404011235","type":"journal-article","created":{"date-parts":[[2004,11,16]],"date-time":"2004-11-16T07:20:18Z","timestamp":1100589618000},"page":"2999-3021","source":"Crossref","is-referenced-by-count":9,"title":["BIFURCATIONS IN APPROXIMATE SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS"],"prefix":"10.1142","volume":"14","author":[{"given":"CHRISTOPHER T. H.","family":"BAKER","sequence":"first","affiliation":[{"name":"Department of Mathematics, The University of Manchester, Manchester, M13 9PL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JUDITH M.","family":"FORD","sequence":"additional","affiliation":[{"name":"Department of Mathematics, UMIST, P.O. Box 88, Manchester, M60 1QD, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"NEVILLE J.","family":"FORD","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University College Chester, Parkgate Road, Chester, CH1 4BJ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0095237","volume-title":"Dynamical Systems","author":"Arnold L.","year":"1995"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/BF03028370"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00476-3"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1112\/S1461157000000322"},{"key":"rf5","first-page":"131","volume":"11","author":"Baker C. T. H.","journal-title":"Electron. Trans. Numer. 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