Table Of ContentInstabilities and Nonequilibrium Structures
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Universita degli Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U. S. S. R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-C. ROTA, M.J. T., Cambridge Mass., U.S.A.
Instabilities and
Nonequilibrium Structures
edited by
Enrique Tirapegui
Departamento de Fisica,
Facultad de Ciencias Fisicas y Matemtiticas,
Universidad de Chile, Santiago, Chile
and
Danilo Villarroel
Departamento de Fisica, Facultad de Ciencias,
Universidad Tecnica Federico Santa Maria,
Valparaiso, Chile
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
Library of Congress Cataloging in Publkation Data
Instabilities and nonequilibrium structures.
(Mathematics and its applications)
"Contains the lectures given at the First International Workshop on Instabilities and
Nonequilibrium Structures which took place in the Universidad Tecnica Federico Santa
Maria, Valparaiso, Chile, December 16-21, 1985"-Frwd.
Includes index.
1. Fluid dynamics--Congresses. 2. Stochastic processes--Congresses. 3. Stabil
ity--Congresses. I. Tirapegui, Enrique. II. Villarroel, D. III. International Work
shop on Instabilities and Nonequilibrium Structures (l~t : 1985 : Universidad Tecnica Federico
Santa Maria) IV. Series: Mathematics and its applications (D. Reidel Publishing Com
pany).
QA911.152 1987 510 86-31392
ISBN-13: 978-94-010-8183-2 e-ISBN-13: 978-94-009-3783-3
DOl: 10.1007/978-94-009-3783-3
Published by D. Reidel Publishing Company,
P.O. Box 17, 3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322,3300 AH Dordrecht, Holland.
All Rights Reserved
© 1987 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover I st edition 1987
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Series Editor's Preface
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perbaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'.
van GuIik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields
does not grow only by putting forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has
changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical
economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the
structure of water meet one another in packing and covering theory; quantum fields, crystal defects
and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering;
and prediction and electrical engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable
systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the
existing classification schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such
(new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical and/or
scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have and have had on
the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of
books which fit the philosophy outlined above. With such books, which are stimulating rather than
definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better
communication among the practitioners in diversified fields.
For some time now mathematics and physics and '" have been increasingly aware that nonlinear
models and phenomena offer opportunities and a richness undreamt of in the linear world. Of
course nonlinear in itself means little and is too understructured to be of use. What we, as scientists,
are currently engaged in, may perhaps be described as isolating a substantial number of kinds of
nonlinear models which can be managed in some way; at the same time the necessary (mathemati
cai) tools and concepts are developed. Certainly the opportunities are fantastic. As I have had
occasion to remark several times before, a purely linear world would be a sad place to live in. And
certainly if our surrounding universe were linear you would not be reading these lines.
Instabilities bifurcations, nonequilibrium structures, and pattern formation, and such related
concepts as chaotic systems and metastability, form a strongly intertwined set of aspects and con
cepts within the general nonlinear field with a well-developed set of mathematical tools, rich in
v
vi SERIES EDITOR'S PREFACE
applications, and also vigorous in generating new concepts and new phenomena demanding under
standing and analysis.
These are what this book is about. And, if the editors and organisors of this workshop and sum
merschool maintain their current high standards in selecting lectures and authors of survey papers, I
have no doubt that further equally interesting volumes on these topics will also appear in this series.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths, their advance was slow
and their applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a beller 'ole, go to it. thenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima
gined.
William Blake
Bussum, December 1986 Michiel Hazewinkel
TABLE OF CONTENTS
FOREWORD ix
LIST OF SPONSORS OF THE WORKSHOP x
LIST OF SUPPORTERS OF THE WORKSHOP x
BRIEF INTRODUCTION TO THE CONTENTS OF THE BOOK
REDUCTION OF THE DYNAMICS OF A BIFURCATION PROBLEM
USING NORMAL FORMS AND SYMMETRIES
G. Iooss 3
STABLE CYCLES WITH COMPLICATED STRUCTURE
J.M. Gambaudo, P. Glendinning and C. Tresser 41
LARGE SCALE INSTABILITIES OF CELLULAR FLOWS
S. Fauve 63
PHYSICAL INTERPRETATION OF OPTICAL BIFURCATIONS
H.lf. Nussenzveig 89
THE INTERACTION OF SOUND AND VORTICITY
F. Lund 117
SPATIO-TEMPORAL INSTABILITIES IN CLOSED AND OPEN
FLOWS
P. Huerre 141
MODELS OF PATTERN FORMATION FROM A SINGULARITY THEO
RY POINT OF VIEW
P. Coullet and D. Repaux 179
PATTERNS AND DEFECTS FAR FROM EQUILIBRIUM
D. Walgraef 197
NONEQUILIBRIUM CORRELATION FUNCTIONS
C.P. Enz 217
FLUCTUATIONS IN UNSTABLE SYSTEMS
N.G. van Kampen 241
WEAK NOISE LIMIT AND NONEQUILIBRIUM POTENTIALS OF
DISSIPATIVE DYNAMICAL SYSTEMS
R. Graham 271
viii TABLE OF CONTENTS
FLUCTUATIONS IN OI}1ETASTABLEOI SYSTEMS
R. Dandoloff and Y. Pomeau 291
NORMAL FORMS WITH NOISE
C. Elphick and E. Tirapegui 299
ADIABATIC NUCLEATION, STABILITY LU!lT AND THE GLASS
TRANSITION TEMPERATURE
E. Meyer 311
MAGNETIC ISLANDS CREATED BY RESONANT HELICAL WIND
INGS
A.S. Fernandes, M.V.A.P. Heller and I.L. Caldas 317
A NON-}~RKOVIAN LORENTZ GAS MODEL
M.O. Caceres 325
SUBJECT INDEX 333
FOREWORD
This book contains the lectures given at the First International
Workshop on Instabilities and Nonequilibrium Structures which
took place in the Universidad Tecnica Federico Santa Maria, Valparaiso,
Chile, from December 16 to 21, 1985.
The idea of this meeting was to give the scientific community
of Chile, and more generally of Latin America, the opportunity
of receiving first hand information on the latest developments
in a field which has known an impressive progress in the last
decade. We think that this is one possible and important way
to help developing countries in their efforts to reduce their
scientific gap and we are very much grateful to all our lecturers
who accepted with great enthusiasm to participate in this Workshop.
It is our intention to organize this meeting every two years
and consequently the second Workshop will take place in December
1987.
We want to thank here our sponsors and supporters whose
interest and help was essential for the success of the meeting.
But the physical realization of the Workshop would have not been
possible without the decisive collaboration of the Universidad
Tecnica Federico Santa Maria. We acknowledge here this support.
We are also especially grateful to Ms. Michele Goldstein, Scientific
Advisor in the French Embassy in Santiago, for her valuable and
constant cooperation.
We are very much indebted to the members of the Departments
of Physics and Mathematics of the Universidad Santa Maria, and
more generally to all the people who helped in the preparation
of the Conference. Ms. Antonina Salazar deserves a special mention
for her assistance in the organization of the meeting and also
for her work in the material preparation of this book. Finally
we express our genuine thanks to Professor Michiel Hazewinkel
for accepting this book in his series and to Dr. David Larner
and the Reidel Publishing Company for their efficient collaboration.
E. Tirapegui
D. Villarroel
LIST OF SPONSORS OF THE WORKSHOP
- Academia Chilena de Ciencias
- Universidad Tecnica Federico Santa Maria
- Minist~re Fran~ais des Affaires Etrang~res
- Proyecto Fortalecimiento del Desarrollo de la Fisica en Chile
(PNUD-UNESCO)
- Departamentos de Investigaci6n y Bibliotecas y de Relaciones
Internacionales de la Universidad de Chile.
- Comisi6n Nacional de Investigaci6n Cientifica y Tecno16gica
(CONICYT)
- Departamento de Fisica de la Universidad de Santiago.
- Sociedad Chilena de Fisica
- Sociedad Chilena de Matematicas
- Centro de Estudios Cientificos de Santiago.
LIST OF SUPPORTERS OF THE WORKSHOP
- Petroquimica Dow S.A.
- Shell Chile S.A.C.E.I.
x
