Bifurcation Problems and their Numerical Solution ISNM 54 International Series of Numerical Mathematics Internationale Schriftenreihe zur Numerischen Mathematik Serie internationale d' Analyse numerique Editors: eh. Blanc, Lausanne; A. Ghizzetti, Roma; R. Glowinski, Paris; G. Golub, Stanford; P. Henrici, Zürich; H. o. Kreiss, Pasadena; A. Ostrowski, Montagnola, and J. Todd, Pasadena Bifurcation Problems and their Numerical Solution Workshop on Bifurcation Problems and their Numerical Solution Dortmund, January 15-17, 1980 Edited by H. D. Mittelmann and H. Weber, Dortmund 1980 Springer Basel AG Llbrary of Congrtlss Cataloging in PubHeation Data Workshop on Bifurcation Problems and their Numerical Solution, University of Dortmund, 1980. Bifurcation problems and their numerical solution. (International series of numerical mathematics ; v. 54) English or German. Bibliography: p. includes index. 1. Differential equations, Nonlinear - Numerical solutions - Congresses. 2. Differential equations, Partial-Numerical solutions - Congresses. 3. Bifurcation theory - Congresses I. Mittelmann, H. D., 1945 - 11. Weber, H., 1948- III. Title IV. Series QA370.W67 1980 515.3'55 80 - 18512 CIP-Kurztitelaufnahme der Deutsdlen BibHothek BHoreation problems and their numerieal solution / Workshop on Bifurcation All rights reserved. No part of Problems and their Numer. Solution, this publication may be reproduced, Dortmund, January 15-17, 1980. stored in a retrieval system, or Ed. by H. D. Mittelmann and H. Weber. - transmitted in any form or by any Basel, Boston, Stuttgart: Birkhäuser, 1980. means, electronic, medlanical, photo (International series of numerical copying, recording or otherwise, mathematics : 54) without the prior permission of the copyright owner. NE: Mittelmann, Hans Detlef [Hrsg.]; Workshop on Bifurcation Problems and ©Springer Basel AG 1980 their Numerical Solution Originally published by Birkhäuser Verlag Basel in 1980. (1980, Dortmund) ISBN 978-3-7643-1204-6 ISBN 978-3-0348-6294-3 (eBook) DOI 10.1007/978-3-0348-6294-3 Contents List of Participants . VI Preface VII Mittelmann, H. D. and Weber, H.: Numerical Methods for Bifurcation Problems - A Survey and Classification . 1 Beyn, W.-I.: On Discretizations of Bifurcation Problems. 46 Bongers, A: Ober ein Rayleigh-Ritz-Verfahren zur Bestimmung kriti- scher Werte . 74 Küpper, T.: Pointwise Error Bounds for the Solutions of Nonlinear Boundary Value Problems 92 Langford, W. F. and Iooss, G.: Interactions of Hopf and Pitchfork Bifurcations . 103 Moore, G. and Spence, A: The Convergence of Approximations to Nonlinear Equations at Simple Turning Points. 135 Scholz, R.: Computation of Turning Points of the Stationary Navier- Stokes Equations Using Mixed Finite Elements. 147 Seydel, R.: Programme zur numerischen Behandlung von Verzwei gungsproblemen bei nichtlinearen Gleichungen und Differential- gleichungen . 163 Voss, H.: Lower Bounds for Critical Parameters in Exothermic Reac- tions 176 Weber, H.: Shooting Methods for Bifurcation Problems in Ordinary Differential Equations . 185 Werner, B.: Turning Points of Branches of Positive Solutions . 211 Wiesweg, U.: Eine numerische Behandlung von primären Bifurkations- zweigen 227 List of Participants Dr. W.-I. Beyn, Universität Konstanz, Fachbereich Mathematik, Postfach 7733, D-7750 Konstanz (FRG) Prof. Dr. R. Böhme, Ruhr-Universität Bochum, Abteilung für Mathematik, Universitätsstraße 150, D-4630 Bochum-Querenburg (FRG) Prof. Dr. E. Bohl, Universität Konstanz, Fachbereich Mathematik, Postfach 7733, D-7750 Konstanz (FRG) Dr. A. Bongers, Universität Mainz, Fachbereich Mathematik, Saarstraße 21, D-6500 Mainz (FRG) Prof. Dr. R. D. Grigorieff, Technische Universität Berlin, Fachbereich Mathematik, Straße des 17. Juni 135, D-I000 Berlin (FRG) Dr. G. Heinemann, GHS Kassel, Fachbereich 17, Heinrich-Plett-Straße 40, D-3500 Kassel Prof. Dr. K. Kirchgässner, Universität Stuttgart, Mathematisches Institut B, Pfaffenwaldring 57, D-7000 Stuttgart 80 (FRG) Dr. T. Küpper, Universität Köln, Mathematisches Institut, Weyertal 86-90, D-5000 Köln 41 (FRG) Prof. Dr. W. F. Langford, McGill University, Department of Mathematics, Montreal, Que. H3A 2K6 (Canada) Prof. Dr. H. D. Mittelmann, Universität Dortmund, Abteilung Mathematik, Postfach 500500, D-4600 Dortmund 50 (FRG) Dr. G. Moore, University of Bath, School of Mathematics, Claverton Down BA2 7AY (U. K.) Prof. Dr. G. H. Pimbley, University of California, Los Alamos Scientific Laboratory, Los Alamos, NM 87544 (USA) Dr. I. Scheurle, Universität Stuttgart, Mathematisches Institut a, Pfaffenwaldring 57, D-7000 Stuttgart 80 (FRG) Dr. R. Scholz, Universität Freiburg, Institut für Angewandte Mathematik, Hermann Herder-Straße 10, D-7800 Freiburg (FRG) Dr. R. Seydel, TU München, Fachbereich Mathematik, Arcisstraße 21, D-8000 München 2 (FRG) Dr. D. Socolescu, Universität Karlsruhe, Institut für Angewandte Mathematik, Englerstraße, D-7500 Karlsruhe (FRG) Prof. Dr. A. Spence, University of Bath, School of Mathematics, Claverton Down BA2 7AY (U. K.) Dr. I. Sprekels, Universität Hamburg, Institut für Angewandte Mathematik, Bundesstraße 55, D-2000 Hamburg 13 (FRG) Prof. Dr. U. Staude, Universität Mainz, Fachbereich Mathematik, Saarstraße 21, D-6500 Mainz (FRG) Or. P. Vielsack, Universität Karlsruhe, Institut für Mechanik, 0-7500 Karlsruhe (FRG) Or. H. Voß, GHS Essen, Fachbereich 6 - Mathematik, Universitätsstraße 2, 0-4300 Essen (FRG) Or. H. Weber, Universität Oortmund, Abteilung Mathematik, Postfach 500500, 0-4600 Oortmund 50 (FRG) Prof. Or. B. Werner, Universität Hamburg, Institut für Angewandte Mathematik, Bundesstraße 55, 0-2000 Hamburg 13 (FRG) Oipl.-Math. U. Wiesweg, Universität Oortmund, Abteilung Wirtschaftswissenschaften, Lehrstuhl für Betriebsinformatik, Postfach 500500, 0-4600 Oortmund 50 (FRG) Preface The >Workshop on Bifurcation Problems and their Numerical Solution<, organized by the editors of this volume, took place at the University of Dort mund from January 15-17, 1980. The main object of this meeting was to bring together mathematicians from different places who were working in this field. Seventeen lectures were presented and discussed at the workshop. Most of them are published in these proceedings together with a survey article on the whole subject. The contributions cover various aspects, ranging from new results in the Ljusternik-Schnirelman theory to numerical methods for the computation of branch points, and from the discretization by multiple shooting to new results about bifurcation mode interactions. Further topics include the non-occur rence of bifurcation for discretized problems and bounds for turning points of nonlinear eigenvalue problems. The organizers would like to express their gratitude to all participants of the workshop, to the authors of the published contributions and especially to the editors of the ISNM-series as well as Birkhäuser Verlag for the rapid publication of these proceedings. Financial support by the >Minister für Wis senschaft und Forschung des Landes Nordrhein-Westfalen< and by the >Ge sellschaft der Freunde der Universität Dortmund< is gratefully acknowledged. Dortmund, May 1980 H. D. Mittelmann H. Weber NUMERICAL METHODS FOR BIFURCATION PROBLEMS - A SURVEY AND CLASSIFICATION H.D. Mittelmann, H. Weber The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory. CONTENTS 1. Introduction 2. Some Applications of Bifurcation 3. Mathematical Tools of Bifurcation Theory 3.1 Analytical Techniques 3.2 Topological Techniques 3.3 Variational Methods 4. General Remarks on the Numerical Solution of Bifurcation ProbLems 5. Numerical Methods 5.1 Discretization Methods 5.2 Numerical Solution of Finite-Dimensional Bifurcation Problems 5.3 Transformation Techniques 5.4 Numerical Determination of Bifurcation Points 5.5 Numerical Solution of Hopf Bifurcation Problems 5.6 Numerical Methods for Turning Point Problems 5.7 Conventional Numerical Methods 5.8 Miscellaneous Results 6. Some Open Problems 7. References 2 Mittelmann/Weber 1. INTRODUCTION Bifurcation theory deals with the analysis of branch points of nonlinear equations in Banach spaces. Let F : X x E + Y be a smooth mapping, where X, Y are Banach spaces and ]I< denotes the field of real or complex numbers. Assume that F(uo' Ao' = O. We are interested in solving the equation (1l F (u,Al o in some neighbourhood of (u,Al = (uo' Ao'. If the Frechet derivative Fu(Uo' Ao' is a homeomorphism from X to Y, then the implicit function theorem assures the existence of a unique smooth branch (u(Al,Al of solutions of (1l: F(u(Al ,Al = 0 with u(Ao' u , defined forlA - A Iless o 0 than some E > o. o Bifurcation theory studies the case where Fu(Uo' Ao' is not invertible. In this singular situation there is a variety of possible sets of solutions of (1l. (uo' Ao' could be a turning point of a solution curve (u(Al,Al of (1l or a simple bifurcation point where two different branches of solutions have a non-tangential intersection. Another possibility is the occurence of a multiple bifurcation point, where more than two branches intersect, see Fig. 1. UfX ~ "Bifurcation diagram"