Pseudospectral approximation of hopf bifurcation for delay differential equations
De Wolff, B. A.J.; Scarabel, F.; Verduyn Lunel, S. M.; Diekmann, O.
(2021) SIAM Journal on Applied Dynamical Systems, volume 20, issue 1, pp. 333 - 370
(Article)
Abstract
Pseudospectral approximation reduces delay differential equations (DDE) to ordinary differential equations (ODE). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyze certain DDE. To substantiate
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the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.
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Keywords: Delay differential equations, Hopf bifurcation, Numerical bifurcation, Pseudospectral method, Taverne, Analysis, Modelling and Simulation
ISSN: 1536-0040
Publisher: Society of Industrial and Applied Mathematics
Note: Funding Information: ∗Received by the editors June 23, 2020; accepted for publication (in revised form) by J. Sieber November 23, 2020; published electronically March 1, 2021. https://doi.org/10.1137/20M1347577 Funding: The work of the first author was supported by the Berlin Mathematical School (BMS). The work of the second author was funded by the NSERC-Sanofi Industrial Research Chair in Vaccine Mathematics, Modeling and Manufacturing. †Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, D - 14195 Berlin (bajdewolff@math.fu-berlin.de). ‡LIAM–Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada; and CD Lab Computational Dynamics Laboratory, Department of Mathematics, Computer Science, and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy (scarabel@yorku.ca). §Department of Mathematics, University of Utrecht, Budapestlaan 6, P.O. Box 80010, 3508 TA Utrecht, The Netherlands (S.M.VerduynLunel@uu.nl, o.diekmann@uu.nl). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.
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