{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T09:27:26Z","timestamp":1686562046277},"reference-count":21,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2003,11]]},"abstract":"<jats:p> We are interested in finding approximate solutions to parameter-dependent Volterra integro-differential equations over long time intervals using numerical schemes. This paper concentrates on changes in qualitative behavior (bifurcations) in the solutions and extends the work of Brunner and Lambert and Matthys (who considered only changes in stability behavior) to consider other bifurcations. We begin by considering a one-parameter equation with fading memory separable convolution kernel: we give an analytical discussion of bifurcations in this case and provide details of the behavior of numerical schemes. We extend our analysis to consider an equation with two-parameter fading memory convolution kernel and show the relationship to the classical test equation studied by the earlier authors. We draw attention to the fact that known stability results may not provide a reliable framework for choice of numerical scheme when other changes in qualitative behavior are also of interest. We give bifurcation plots for a variety of methods and show how, for known values of the parameters, stepsizes h&gt;0 may be chosen to preserve the correct qualitative behavior in the numerical solution of the Volterra integro-differential equation. <\/jats:p>","DOI":"10.1142\/s0218127403008570","type":"journal-article","created":{"date-parts":[[2003,12,18]],"date-time":"2003-12-18T09:02:05Z","timestamp":1071738125000},"page":"3255-3271","source":"Crossref","is-referenced-by-count":4,"title":["BIFURCATIONS IN NUMERICAL METHODS FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS"],"prefix":"10.1142","volume":"13","author":[{"given":"JOHN T.","family":"EDWARDS","sequence":"first","affiliation":[{"name":"Department of Mathematics, University College Chester, Parkgate Road, Chester, CH1 4BJ, 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"}]},{"given":"JASON A.","family":"ROBERTS","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":"publisher","DOI":"10.1137\/0716066"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-6294-3_2"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1137\/0724072"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01933399"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02239501"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/0771-050X(82)90044-4"},{"key":"rf7","volume-title":"The Numerical Solution of Volterra Equations","author":"Brunner H.","year":"1986"},{"key":"rf8","first-page":"55","volume":"6","author":"Edwards J. 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