Table Of ContentISNM70:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serle internationale d'A nalyse numerique
Vol. 70
Edited by
Ch. Blanc, Lausanne; A. Ghizzetti, Roma;
R. Glowinski, Paris; G. Golub, Stanford;
P. Henrici, Ziirich; H. 0. Kreiss, Pasadena;
A. Ostrowski, Montagnola; J. Todd, Pasadena
Springer Basel AG
Numerical Methods for
Bifurcation Problems
Proceedings of the Conference at the University of Dortmund,
August 22-26, 1983
Edited by
T. Kiipper
H. D. Mittelmann
H. Weber
Springer Basel AG 1984
Editors
T.KUpper H. Weber
H. D. Mittelmann J ohannes Gutenberg-Universitat
Universitat Dortmund Rechenzentrum
Abt. Mathematik Postfach 3980
Postfach 50 05 00 D-6500 Mainz 1
D-4600 Dortmund 50
Library of Congress Cataloging in Publication Data
Numerica! methods for bifurcation problems.
(International series of numerica} mathematics; v. 70)
1. Numerica} analysis-Congresses. 2. Bifurcation
theory-Congresses. 1. KUpper, T. (Tassilo), 1947-
11. Mittelmann, H. D., 1945- , III. Weber, H.,
1948- . IV. Series.
QA297.N864 1984 519.4 84-9286
ISBN 978-3-0348-6257-8 ISBN 978-3-0348-6256-1 (eBook)
DOI 10.1007/978-3-0348-6256-1
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Numerica! methods for bifurcation problems :
proceedings of the conference at the Univ. of
Dortmund, August 22-26, 1983 1 ed. by
T.KUpper ...
(International series of numerical mathe
matics; 70)
ISBN 978-3-0348-6257-8
NE: KUpper, Tassilo [Hrsg.]; Universitat
(Dortmund); GT
Ali rights reserved.
No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior
permission of the copyright owner.
© 1984 Springer Basel AG
Originally published by Birkhăuser Verlag Basel in 1984
Softcover reprint ofthe hardcover lst edition 1984
ISBN 978-3-0348-6257-8
5
FOreword
These are the Proceedings of the Conference on
Numerical Methods for Bifurcation Problems
which was held August 22 - 26, 1983 at the University of Dortmund as a sequel
to _a smaller Workshop held at Dortmund in 198o. The conference was organized
in connection with the GAMM committees Diskretisierende Methoden in der Fest
korpermechanik and Effiziente numerische Verfahren fur partielle Differential
gleichungen.
The aim of this conference was to provide an up to date survey on the current
state of the art in applying numerical methods to bifurcation problems as
well as to inform about most recent developments and, last not least, to
bring together numerical analysts and engineers working in the field. For
that reason papers of both expository and research character were welcome.
The published articles have been subject to a reviewing process.
The majority of the participants were numerical analysts, and the volume
reflects this fact. A number of articles, however, deal with problems of
direct relevance to the applied sciences, in particular to problems in bio
logy and engineering.
It is a pleasure for us to thank several agencies for their help. The confe
rence was mainly supported by a generous grant of the Deutsche Forschungsge
meinschaft. Participation of colleagues from Eastern European countries was
made possible through a special program of the Deutscher Akademischer Aus
tauschdienst. A donation of the Freundesgesellschaft der Universitat Dortmund
helped us to cover organizational costs. We further acknowledge the hospita
lity of the University of Dortmund and we like express our sincerest thanks
to Mrs. u. Krieglstein and Mrs. I. Speck for their helpful assistance. Fi
nally we greatly appreciate the rapid publication of these proceedings and
we express our gratitude to Birkhauser-Verlag and to the editors of the
ISNM-series.
December 1983 Tassilo Kupper
Hans Detlef Mittelmann
Helmut Weber
7
INDEX
5
For~rd. ••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
List of Participants •••.••••••..••••••••••••••••••••••••.•.•••••• 11
~.E.:
Bifurcations Arising in the Calculation of Critical Points
via lbnc>t<'l'}' ~tllc:xis ...•......•..........•................... 15
BERNARDI, <ll. RAPPAZ, J.
I
Approximation of Hop£ Bifurcation for Semilinear Parabolic
Fq\lat iOI'lS •••••••••••••••••••••••••••••••••••••••••••••••••••• 29
BEYN, W.-J.:
Defining Equations for Singular Solutions and Numerical
AJ?plicatiC>l'ls ••••••••••••••••••••••••••••••••••••••••••••••••• 42
BEYN, w. -J. BCHL, E.:
I
Organizing Centers for Discrete Reaction Diffusion Models •••• 57
BCHL, E.:
Discrete Versus Continuous Models for Dissipative Systems •••• 68
BRE2ZI, F., USHIKI, S., FUJII, H.:
"Real" and "Ghost" Bifurcation Dynamics in Differen:::e
Scl\elnes for OOEs ••••••••••••••••••••••••••••••••••••••••••••• 79
CALUolAERl'S R. :
I
The Use of Extrapolation for the Solution of Bifurcation
Problems. • . . . • . • • . • . • . • • • • • . • . • • . . • . • • . • • • • • . . • • . • . . . . . . • • . • . 10 5
CF..AN, T.F.:
Techniques fcc Large Sparse Systems Arising from Continuation
Methods • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 116
CLIFFE, K.A. SPDO:, A.
I
The Calculation of High Order Singularities in the Finite
Taylor Pr'oblem............................................... 129
DESCI.aJX, J. :
On Hqpf and Subharmonic Bifurcations......................... 145
GRIB'lANK, A.:
Quadratically AJ?pended Linear Models for Locating Generalized
Tllrni113 l'c>ints. . . . . . . . . • . . . . . • . . . . . . . . . . . . . . . . • . . . . . . . . • . . . . . 162
HADELER, K. P. :
Hysteresis in a Model for Parasitic Infection................ 171
8
HOLODNIOK, M. KUBICEK, M.
I
Continuation of Periodic Solutions in Ordinary Differential
Fquations - Numerical Algorithm and Application to lorenz
181
~el........................................................
JEPSON, A.D. SPENCE, A.:
I
Singular POints and their Oomputation........................ 195
KEARFOIT, R.B.:
On a General Technique for Finding Directions Proceeding
from Bifurcation POints...................................... 210
KELLER, H. B. JEPSeN, A.D.
I I:
Steady State and Periodic Solution Paths: their Bifurcations
am Q::arpi tat ions ................................•............
KUBICEK M. HOIDDNJOK, M.
I
Numerical Determination of Bifurcation POints in Steady State
and Periodic Solutions- Numerical Algorithms and Examples ••• 247
KUEPPER, T. KUSZTA, B. :
I
Feedback Stimulated Bifurcation• ••••••••••••••••••••••••••••• 271
LANGFORD, W. F. :
285
Numerical Studies of Tbrus Bifurcations ••••••••••••••••••••••
w.
MACKENS, JARAUSCH, H.
I I:
Numerical Treatment of Bifurcation Branches by M.aptive
Olrlciensat ion ................................................ . 296
MENZEL, R.:
Numerical Determinaticn of Multiple Bifurcation POints ••••••• 310
MITI'EU1ANN, H. D. I:
Continuation Near Symmetry-Breaking Bifurcation POints ••••••• 319
KX>RE, G.:
The Numerical BuCkling of a Visco-elastic Rod •••••••••••••••• 335
t.uNZ, H.:
Asymptotic Error Expansion for Finite Difference Sdhemes
for Elliptic Systems Near Turning POints• •••••••••••••••••••• 344
PEI'roEN, H.-o. PRUEFER, M.:
I
Global Aspects of Newton's Method for Nonlinear Boundary
352
Value ProbleJD.s ••.••••••••.•..•.••••.•.•••••.•••..•••••••••••.
RAU3EL, G. GEYMCNAT, G.:
I
Finite Dimensional Approximation of Some Bifurcation Problems
in Presence of Symmetries.................................... 369
w.
REDDIEN G. GRIEWANK, A. :
I I
Computation of Generalized Turning POints and Two-POint
Boundary Value ~!ems...................................... 385
9
RHEINBOLDI' w. c. :
I
On Some Methods for the Computational Analysis of Manifolds.. 401
ROOSE, D. CAUJWAERI'S R. :
I I
Direct Methods for the Computation of a ~bnsinple Turning
Point Cbrre~nding to a cu~................................ 426
SCHEIDL, R.
On the Axisymmeteric Buckling of Thin Spherical Shells....... 441
SCHOLZ: R.:
On the Rate of Convergence for the Approximation of Nonlinear
Problems..................................................... 452
SCH'lETLI<l<, H.:
Algorithms for Finite-Dimensional Turning Point Problems from
Viewpoint to Relationships with Constrained Optimization
Methods • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 4 59
SEYDEL, R.:
A Continuation Algorithm with Step Control................... 480
SIJBRAND, J. DIPRIMA, R.C. FAGLES, P.M.:
I I
Bifurcation Near Multiple Eigenvalues for the Flow Between
Concentric Counterrotating Cylinders......................... 495
SPFN:E, A. JEPSeN, A.D.:
I
The Numerical Calculation of Cusps, Bifurcation Points and
Isola Formation Points in Two Parameter Problems............. 502
STEINRUEX::K, H. I TROOER, H. I \~ISS, R. :
Mode Jumping of Imperfect, Buckled, Rectangular Plates....... 515
TRCGER, H.:
Application of Bifurcation Theory to the Solution of
Nonlinear Stability Problems in Mechanical Engineering....... 525
WEBER, H.:
A Singular Multi-Grid Iteration Method for Bifurcation
ProbleJtls. . . . • . . • . . . • . • • . • . . • . • . . • . . . . • • • . . . • • . . . • . • • • . . . . . . • • 54 7
WERNER, B.:
Regular Systems for Bifurcation Points with Underlying
SylnlnE!tries.............................................. . . . . . 562
rJEYER. J. , ASI'ABURUIIGA, M.A. , FIGUEROll., J. :
Newton Iterates for Positive Solutions of a Class of
Nonlinear Eigenvalue Problems................................ 575
11
List of Participants
Allgower, E. Dept. of Math., Colorado State Univ.,
Fort Collins, Colorado 80523
Bernardi, Christine C.N.R.S. and University P. et M. Curie,
Analyse Numerique
Universite P. et M. Curie, 4 Place jussieu
75230 Paris, Cedex 05, France
Beyn, Wolf-JUrgen Fakultat fUr Mathematik, Universitat Konstanz
Postfach 5560, D-7750 Konstanz
Bischoff, Dieter Univ. Hannover, I.f. Baumech. u. Numer. Mech.
Callinstr. 32, 3000 Hannover 1
Bohl, Erich Universitat Konstanz, Fakultat fUr Mathematik
Postfach 5560, 7750 Konstanz
Belley, Catherine I.N.S.A. Laboratorie d'Analyse Numerique
20 Av. des Bultes de Coesmes
35043 Rennes Cedex, France
Bersch-Supan, W. FB 17 Mathematik, Joh. Gutenberg-Univ. Mainz
Postfach 3980, 6500 Mainz
Braess, Dietrich Institut fUr Mathematik, Ruhr-Universitat Bochum
Universitatsstr. 150, Geb. NA, 463 Bochum
Brezzi, Franco Dept. of Structural Mechanics and I.A.N. of C.N.R.
Universita di Pavia, C.S.O. Strada Nueva 65
27100 Pavia, Italy
Brown, Adrian University of Bath, School of Mathematics
Claverton Down, Bath BA2 7AY, England
Caluw~arts, Renaat Katholieke Universiteit Leuven
Dept. Computerwetenschappen
CelestUnenlaan 200 A, 3030 Heverlee, Belgien
di Carlo, Antonio Universiti di Roma "La Sapienza"
Istituto di Scienza delle Costruzioni,
Facolta di Ingegneria, Via Eudossiana 18,
00184 Roma, Italy
Chan, Tony F. Yale University, Computer Science Dept.
Box 2158, Yale Station,
New Haven, CT 06520, USA
Cliffe, K.A. Aere Harwell, Theoretical Physics Division
Oxfordshire, OXII ORA
Descloux, J. EPFL Lausanne, Dept. Mathematik
MA (Eeublens), CH-1015 Lausanne
Doring, B. Mathemntisches Institut der Universitat DUsseldorf
Universitatsstr. 1, 4 Dusseldorf 1
Gbthel, Rainer Mathe rna ti schcs I nstitut
Universitiit Dortmund
Greenway, Philip University of Bath, School of Hathematies
Claverton Down, Bath, England
Griewank, Andreas Dept. of Mathematics SMV
Southern Methodist Unlversity
Dallas, Tx 75275 USA
