The IMA Volumes in Mathematics

and its Applications

Volume 119

Series Editor Willard Miller, Jr.

Springer Science+Business Media, LLC

Institute for Mathematics and its Applications

IMA

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Continued at the back

Eusebius Doedel Laurette S. Tuckerman Editors

Numerical Methods for Bifurcation Problems

and Large-Scale Dynamical Systems

With 129 Illustrations

Springer

Eusebius Doedel Laurette S. Tuckerman Laboratoire d'Informatique pour la Applied Math 217-50

California Institute of Technology Pasadena, CA 91125, USA

Mecanique et les Sciences de 1'Ingenieur (LIMSI) B.P. 133 91403 Orsay, France

Series Editor: Willard Miller, Jr. Institute for Mathematics and its

Applications University of Minnesota Minneapolis, MN 55455, USA

Mathematics Subject Classification (1991): 34-04, 34-06, 35-04, 35-06, 65L99, 65M99, 65N99

Library of Congress Cataloging-in-Publication Data Doedel, Eusebius.

Numerical methods for bifurcation problems and large-scale dynamical systems / Eusebius Doedel, Laurette S. Tuckerman.

p. cm — (The IMA volumes in mathematics and its applications ; 119) Includes bibliographical references and index. ISBN 978-1-4612-7044-7 ISBN 978-1-4612-1208-9 (eBook) DOI 10.1007/978-1-4612-1208-9 1. Differential equations—Numerical solutions. 2. Bifurcation theory. 3. Differentiate

dynamical systems. I. Tuckerman, Laurette S. II. Title. III. IMA volumes in mathematics and its applications ; v. 119. QA377.D655 2000

515'.35—dc21 99-088039

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FOREWORD

This IMA Volume in Mathematics and its Applications

NUMERICAL METHODS FOR BIFURCATION PROBLEMS AND LARGE-SCALE DYNAMICAL SYSTEMS

is based on the combined proceedings of two workshops devoted to compu­tational issues.

The workshops were an integral part of the 1997-98 IMA program on "EMERGING APPLICATIONS OF DYNAMICAL SYSTEMS."

I would like to thank Donald G. Aronson, University of Minnesota (Mathematics); Wolf-Juergen Beyn, Universitaet Bielefeld (Fakultaet fuer Mathematik); Eusebius Doedel, California Institute of Technology (Applied Mathematics); Bernold Fiedler, Free University of Berlin (Mathematics); H.B. Keller, Caltech (Applied Mathematics); Yannis Kevrekidis, Princeton University (Chemical Enginering); Jens Lorenz, University of New Mex­ico (Mathematics and Statistics); Edriss S. Titi, University of California (Mathematics); Laurette S. Tuckerman, Laboratoire d'Informatique pour la Mecanique et les Sciences de l'Ingenieur (LIMSI) for their excellent work as organizers of the meeting.

Special appreciation to Eusebius Doedel and Laurette S. Tuckerman for serving as editors of the proceedings.

I also take this opportunity to thank the National Science Foundation (NSF), and the National Security Agency (NSA), whose financial support made the workshop possible.

Willard Miller, Jr., Professor and Director

v

PREFACE

The papers in this volume are based on lectures given at the first two workshops held as part of the 1997-1998 IMA Academic Year on Emerging Applications of Dynamical Systems. This IMA Year was organized by John Guckenheimer (chair), Eusebius Doedel, Martin Golubitsky, Yannis Kevrekidis, Rafael de La Llave, and John Rinzel. The scientific program had a strong computational component, as especially reflected in the first two workshops, which were entirely devoted to computational issues.

Workshop 1, "Numerical Methods for Bifurcation Problems," was held in the week of September 15-19, 1997. The organizing committee of this workshop consisted of Eusebius Doedel (chair), Wolf-Juergen Beyn, Ber­nold Fiedler, Yannis Kevrekidis, and Jens Lorenz. The workshop con­centrated on complex computational issues in dynamical systems. While computational techniques for low-co dimension local bifurcations in few­degree of freedom systems are in advanced state of development, much work remains to be done on the numerical treatment of higher codimension singularities. More importantly, there is a pressing need for the develop­ment of numerical methods for computing global objects in phase space, their interactions and bifurcations. This workshop brought together math­ematicians, numerical analysts, and computer scientists working on these problems. Particular topics included the detection of bifurcations and the development of associated numerical and visualization software. Also con­sidered were important theoretical issues, such as smooth factorization of matrices, self-organized criticality, and singular heteroclinic cycles. The numerical computation of manifolds, such as invariant tori and resonance surfaces were also studied.

Workshop 2, "Large Scale Dynamical Systems," was held during the week of September 29-0ctober 3, 1997. It was organized by Laurette Tuckerman (chair), Edriss Titi, Herbert Keller, and Don Aronson. The numerical study of low-dimensional dynamics in large scale sets of ODEs and discretizations of PDEs necessitates the development of special pur­pose algorithms for simulations, stability and bifurcation analysis. This workshop addressed the development and application of special iterative methods for large scale systems. It also considered global model reduction schemes for PDEs. A related goal is to encourage the interpretation of large-scale physical problems as dynamical systems which, although high­dimensional, undergo low-codimension bifurcations. Applications of special interest include selected problems arising in fluid flows, and pattern forma­tion in reaction-diffusion systems.

vii

viii PREFACE

We would like to thank the IMA and the program coordinators for holding this workshop. We thank outgoing and incoming directors A vner Friedman and Willard Miller, and especially Robert Gulliver for coordinat­ing the workshops, and the IMA staff for providing logistic support. We also thank Patricia V. Brick for her important contribution to this volume as editorial and production coordinator at the IMA.

Eusebius Doedel Applied Math 217-50 California Institute of Technology

Laurette S. Tuckerman Laboratoire d'Informatique pour la Mecanique et les Sciences de l'Ingenieur (LIMSI)

August 1999

CONTENTS

Foreword ............................................................. v

Preface ............................................................. vii

Numerical bifurcation techniques for chemical reactor problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1

V. Balakotaiah and J. Khinast

Path-following of large bifurcation problems with iterative methods .................................................... 37

K. Bohmer, Z. Mei, A. Schwarzer, and R. Sebastian

On the bifurcation from continuous to segmented chip formation in metal cutting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67

T.J. Burns, M.A. Davies, and C.J. Evans

Using dynamical system tools in Matlab .............................. 85 Won Gyu Choe and J. Guckenheimer

Formation and instabilities of coherent structures in channel flows ..................................................... 115

Katie Coughlin

Applications of smooth orthogonal factorizations of matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141

Luca Dieci and Timo Eirola

Continuation of codimension-2 equilibrium bifurcations in CONTENT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163

W. Govaerts, Yu. A. Kuznetsov, and B. Sijnave

Inclination-flips in the unfolding of a singular heteroclinic cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 185

Ale Jan Homburg

Investigating torus bifurcations in the forced Van der Pol oscillator ............................................... 199

Bernd Krauskopf and Hinke M. Osinga

ix

x CONTENTS

Quasiperiodic response to parametric excitations. . . . . . . . . . . . . . . . . . .. 209 J.M. Lopez and F. Marques

Self-organized criticality: analysis and simulation of a 1D sandpile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229

Jens Lorenz, Steven Jackett, and Wangguo Qin

Computation and bifurcation analysis of periodic solutions of large-scale systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 265

Kurt Lust and Dirk Roose

Multiple equilibria and stability of the north-atlantic wind-driven ocean circulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 303

M. Jeroen Molemaker and Henk A. Dijkstra

Numerical exploration of bifurcation phenomena associated with complex instability .................................. 319

Merce Olle

Chaos in traveling waves of lattice systems of unbounded media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 327

D.R. Orendovici and Ya.B. Pesin

Pattern formation in a cellular slime mold ........................... 359 H. G. Othmer, B. Lilly, and J. C. Dalton

Global parametrization and computation of resonance surfaces for periodically forced oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 385

Bruce B. Peckham

Computing invariant tori and circles in dynamical systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 407

Volker Reichelt

A Design problem for image processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 439 R. Seydel

Bifurcation analysis for timesteppers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 453 Laurette S. Tuckerman and Dwight Barkley

List of participants ................................................. 467