Table Of ContentInstabilities and Nonequilibrium Structures HI
Mathematics and Its Applications
Managing Editor:
M HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Universitä degli Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
M. NIVAT, Universiti de Paris VII, Paris, France
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G. -C. ROTA, MJ.T., Cambridge, Mass., USA.
Volume 64
Instabilitie s and
Nonequilibriu m
Structures II I
edited by
E. Tirapegui
Facultad de Ciencias Fisicas y Matemdticas,
Universidad de Chile,
Santiago ,Chile
and
W. Zeller
Institute* de Fisica,
Universidad Catölica de Valparaiso,
Valparaiso, Chile
*
SPRINGER SCIENCE+BUSINESS MEDIA , B.V.
Library of Congress Cataloging-in-Publication Data
Instabilitie s and nonequ11Ibr1un structure s III / edite d by E.
T1rapegu1 and W. Zeller .
p. cm. — (Mathematics and It s applications )
Includes Index.
ISBN 978-94-010-5522-2 ISBN 978-94-011-3442-2 (eBook)
DOI 10.1007/978-94-011-3442-2
1. Flui d dynamics—Congresses. 2. Stochasti c processes-
-Congresses. 3. Stabl1 ity—Congresses . I. Tlrapegul , Enrique.
II. Zeller , W. III. Title : Instabilitie s and nonequ111br1um
structure s three . IV. Series : Mathematics and It s application s
(Kluwer Academic Publishers )
QA911.1532 1991
532—dc20 91-18950
IISSBBNN 997788--9944-0-0110-05-552522-222-
Printed on acid-free paper
All Rights Reserved
©1991 Springer Science+Busines sMedia Dordrecht
Originally published by Kluwer Academci Publishesr in 1991
Softcover reprint of the hardcover 1st edition 1991
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanica,l
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
'Ht moi, ...• Ii j'avait so comment en revenir, One service mathematics has rendered the
je ny _ais point aile':' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf neJll
to the dusty canister labelled 'discarded non
The series is diwrgent; therefore we may be sense'.
• ble to do something with it. Eric T. Bell
O. He niside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, alI kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d't!tre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"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."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can hclp mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radarl sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vi SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. AU three have applications in both
electrical engineering and physics. Once, complex numbers were equaUy outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization 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.
How can one control instabilities and nonequilibrium structures, or, better, make good use of them.
Nature does so quite effectively and on a large scale; so do aU kinds of athletes; by and large tech
nology does not. The first steps consist in understanding and describing such structures and that is
a branch of mathematics that is in fun swing right now as this volume will testify. I called it a
branch of mathematics in the previous line but actuany it is more like a web of interconnections
linking very many concepts and ideas: commutative and noncommutative (quantum) probability,
universality in chaos, topological solitons, cellular automata, fragmentation, changes in symmetry,
bifurcations, pattern formation, renormalization, ... This list but a few of the characteristic phrases
and aU of them are discussed in the present volume which thus represents a rich source of material
from the unstable nonequilibrium world (which is just like the one we live in; fortunately for us).
The monest patb between t_ trutb. in tbe Never lend books, for no one ever returns
real domain passes tbrough tbe complex tbem; the only book. I have in my library
domain. are book. tbat oIber folk bave lent me.
J. Hadamard Anatole France
La pbysique ne nous dcnne paa _Iement The function of an expert is not to be more
l'occasion de re'Ioudre des probltlmes ... cDe riabt than other pccpJe, but to be wrona for
noul fait pressentir la solution. more sopbisticated reasons.
H. Poincare David Butler
Bussum, June 1991 Michiel Hazewinkel
FOREWORD
We present here a selection of the lectures given at the Third Interna
tional Workshop on Instabilities and Nonequilibrium Structures in Valpa
raiso, Chile, in December 1989. The Workshop was organized by Facultad
de Ciencias Fisicas y Matematicas of Universidad de Chile and by
Instit~
to de Fisica of Universidad Cat6lica de Valparaiso where it took place.
This periodic meeting takes place every two years in Chile and aims
to contribute to the efforts in Latin America towards the development of
scientific research. This development is certainly a necessary condition
for progress in our countries and we thank our lectures for their warm
collaboration to fulfill this need. We are also very much indebted to
the Chilean Academy of Sciences for sponsoring officially this Workshop.
We thank also our sponsors and supporters for their valuable help,
and most especially UNESCO, National Science Foundation (USA), and Fun~
ci6n Andes of Chile. The efforts of M.Alain Siberchicot, Scientific Ad
visor at the French Ambassy in Santiago, have been essential for our
su~
cess and we acknowledge here the generous support of the Scientific Coo
peration Program of France for Chile. Ms. Alda Bertoni and Cecilia Cam
pos deserve a special mention for their remarkable work during the reall
zation of the Workshop. We acknowledge also the help of Ms. Elsa Nanco
in the material preparation of this book. We are grateful to Professor
Michiel Hazewinkel for including this book in his series and to Dr. Da
vid Larner of Kluwer for this continuous interest and support to this
project.
E. Ti rapegui
w. Zeller
vii
LIST OF SPONSORS OF TI£ WORKSfDP
- Academia Chilena de Ciencias
- Academia de Ciencias de America Latina
- Facultad de Ciencias Fisicas y Matematicas de la Universidad de Chile
- Instituto de Fisica de la Universidad Catolica de Valparaiso
- CONICYT (Chile)
- Ministere Francais des Affaires Etrangeres
- lNESCO
- International Centre for Theoretical Physics (Trieste)
- DFG (Gennany)
- FNRS (Belgilm)
- Fundacion Andes (Chile)
- Sociedad Chilena de Fisica
- Departamento Tecnico de Investigacion y de Relaciones Internacionales
de la Universidad de Chile
viii
TABLE OF CONTENTS
FOREWORD vii
LIST OF SPONSORS OF TI£ WORKSI{)P vi i i
PREFACE 1
PART I. DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS 3
FAST DYNAMO FOR SOME ANOSOV FLOWS
P. Collet 5
J()W I{)RSESI{)ES ARE CREATED
J.M.Gambaudo and C.Tresser 13
FINE STRUCTURE OF UNIVERSAL CANTOR SETS
C.Tresser 27
HAMILTONIAN CHAOS IN A LASER DAMAGE MODEL
W.Becker. M.Fuka. J.K.McIver. M. Orszag and R. Ramirez 43
PERIODIC MODULATION OF A LOGISTIC OSCILLATOR
M.Kiwi. E.Lazo and J.Rossler 59
QUI\NTLM SPIN CHAINS WITH RESIDUAL ENTROPY
B.Nachtergaele 65
QUl\NTLM FLUCTUI\TIONS AND LINEAR RESPONSE THEORY
A.Verbeure 75
NON-COMMUTATIVE LARGE DEVIATIONS AND APPLICATIONS
G.A.Raggio 81
SQUEEZING IN THE SU(1.1) GROUP
J.Aliaga. G.Crespo and A.N.Proto 87
FULL SCALING AND REAL-SPACE RENORMALIZATION
P.Cordero 95
SAND PILES. COMBINATORIAL GAMES AND CELLULAR AUTOMATA
E. Goles 101
x TABLE OF CONTENTS
PART II. STOCHASTIC EFFECTS ON DYNAMICAL SYSTEMS 123
NONEQUILIBRIUM POTENTIALS IN SPATIALLY EXTENDED PATTERN
FORMING SYSTEMS
R.Graham and T.Tel 125
PASSAGE TIME DESCRIPTION OF DYNAMICAL PROCESSES
M.San Miguel, E.Hernandez-Garcia, P.Colet, M.O.Caceres and
F.De Pasquale 143
STOCHASTIC PROCESSES DRIVEN BY COLORED NOISE: A PATH INTEGRAL
POINT OF VI EW
H.S.Wio, P.Colet, M.San Miguel, L.Pesquera and M.A.Rodrlguez 157
EFFECT OF NOISE ON A HQPF BIFURCATION
R.C.Buceta. E.Tirapegui 171
SOME PROPERTIES OF QUASI STATIONARY DISTRIBUTIONS IN THE BIRTH
AND DEATH CHAINS: A DYNAMICAL APPROACH
P.A.Ferrari, S.Martinez and P.Picco 177
PI()TON NOISE REDLtTION IN LIGHT SOURCES
M.Orszag and J.C.Retamal 189
PART III. INSTABILITIES IN NONEQUILIBRIUM SYSTEMS 199
SPATIO-TEMPORAL PROPERTIES OF CENTRIFUGAL INSTABILITIES
I.Mutabazi and J.E.Wesfreid 201
SHAPE OF STATIONARY AND TRAVELLING CELLS IN THE PRINTER'S
INSTABILITY
M.Rabaud and V.Hakim 217
SECONDARY INSTABILITY OF CELLULAR STRUCTURES IN DIRECTIONAL
SOLIDI FICATION
A. Karma 225
VELOCITY FIELD STRUCTURE AND SEMIQUANTITATIVE ANALYSIS OF
TRACER DISPERSION IN A TAYLOR-VORTEX FLOW OF WIDE GAP
A.Barrantes, A.Calvo and M.Rosen 233
LASERS AS A TEST BENCH FOR THEORIES OF NON-EQUILIBRIUM
STRLtTURES
J.R.Tredicce, E.J.D'Angelo, C.Green, G.B.Mindlin, L.M.Narducci,
G.L.Oppo and H.Solari 239
