Nonlinear Evolution and Chaotic Phenomena

332 Pages·1988·10.13 MB·English

by  Giovanni Gallavotti (auth.)

Nonlinear Evolution
and Chaotic
Phenomena
NA TO ASI Series
Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The series is published by an international board of publishers in conjunction with the
NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics New York and London
C Mathematical Kluwer Academic Publishers
and Physical Sciences Dordrecht, Boston, and London
D Behavioral and Social Sciences
E Applied Sciences
F computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York, London,
H Cell Biology Paris, and Tokyo
Recent Volumes in this Series
Volume 170-Physics and Applications of Quantum Wells and Superlattices
edited by E. E. Mendez and K. von Klitzing
Volume 171-Atomic and Molecular Processes with Short Intense Laser Pulses
edited by Andre D. Bandrauk
Volume 172-Chemical Physics of Intercalation
edited by A. P. Legrand and S. Flandrois
Volume 173-Particle Physics: Cargese 1987
edited by Maurice Levy, Jean-Louis Basdevant, Maurice Jacob,
David Speiser, Jacques Weyers, and Raymond Gastmans
Volume 174-Physicochemical Hydrodynamics: Interfacial Phenomena
edited by Manuel G. Velarde
Volume 175-Superstrings
edited by Peter G. O. Freund and K. T. Mahanthappa
Volume 176-Nonlinear Evolution and Chaotic Phenomena
edited by Giovanni Gallavotti and Paul F. Zweifel
Series B: Physics
Nonlinear Evolution
and Chaotic
Phenomena
Edited by
Giovanni Gallavotti
Centro Linceo Interdisciplinare "B. Segre"
Accademia dei Lincei
Rome, Italy
and
Paul F. Zweifel
Virginia Polytechnic Institute and State University
Blacksburg, Virginia
Plenum Press
New York and London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of a NATO Advanced Study Institute on
Nonlinear Evolution,
held June 8-19,1987,
in Noto, Sicily, Italy
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute on Nonlinear Evolution (1987: Noto, Sicily)
Nonlinear evolution and chaotic phenomena I edited by Giovanni Gallavotti
and Paul F. Zweifel.
p. cm.-(NATO ASI series. Series B, Physics; v. 176)
"Proceedings of a NATO Advanced Study Institute on Nonlinear Evolution held
June 8-19,1987, in Noto, Sicily, Italy"-T.p. verso.
"Published in cooperation with NATO Scientific Affairs Division.
Includes bibliographical references and index.
ISBN-13:978-1-4612-8294-5 e-ISBN-13:978-1-4613-1017-4
001: 10.1007/978-1-4613-1017-4
1. Chaotic behavior in systems-Congresses. 2. Particles (Nuclear physics)
Congresses. 3. Statistical mechanics-Congresses. I. Gal Ia votti , Giovanni. II.
Zweifel, P. F. (Paul Frederick), 1929- III. North Atlantic Treaty Organization.
Scientific Affairs Division. IV. Title. V. Series.
Q172.5.C45N38 1987 88-9415
530.1-dc19 CIP
© 1988 Plenum Press, New York
Softcover reprint of the hardcover 18t edition 1988
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying,
microfilming, recording, or otherwise, without written permission from the Publisher
PREFACE
This volume represents the proceedings of a NATO Advanced Study
Institute held at Noto, Sicily June 8-19, 1987. The director was
Giovanni Gallavotti, Roma, with co-directors Marcello Anile, Catania
and P. F. Zweifel, Virginia Tech. Other members of the scientific
organizing committee included Mitchell Feigenbaum, Rockefeller
University and David Ruelle, IHES.
The attendance at the school consisted of 23 invited speakers and
approximately 80 "students", the term student being in quotation marks
because many of them were of post-doctoral or even professorial status,
although there were also a goodly number of actual graduate students in
attendance also. Because of the disparate background of these
"students", it was felt advisable to include at the conference special
tutorials each afternoon, in which the contents of the morning's
lectures were reviewed and clarified as necessary. These tutorials,
organized by Gallavotti, involved various of the speakers, organizers,
and other senior members of the school, and contributed in no little way
to the overall success of the school. The organizers of the school
would like to take this opportunity to thank all of those who assisted
in these sessions, and to assure them that the results were definitely
worth the effort.
Also contributing to the success of the school were a number of
contributed papers, presented during the course of the afternoon
tutorials. Three of those papers are included in these proceedings;
they are the papers of DiFrancesco; Gallimbeni, Miari and Sertorio
(presented by Sertorio); and Vittot. Again, the organizers would like
to express their thanks to all those who gave contributed papers, and
especially to those who supplied manuscripts. (A listing of the
contributed papers for which manuscripts were not received appears at
the end of the proceedings.)
Of the 23 invited speakers, three (McCormick, Libchaber, and
Franceschini) never supplied manuscripts. After waiting for six
months, the editors have decided not to delay publication any further.
Summaries of the missing papers (prepared by the author of this preface
are included in the proceedings.
The paper by Koch and Wittwer was presented by Wittwer; that of
Krug and Spohn by Spohn and that of Keating and Mondragon by Michael
Berry, also of Bristol. In point of fact, Berry described his own
work--Keating and Mondragon acted as recorders. In a similar way, the
talks of Lanford were written up by Sam Robinson, University of North
Carolina-Wilmington; those of Feigenbaum by Edmond Rusjan, Virginia
Tech; and those of Swinney by P. F. Zweifel.
v
We have attempted to arrange the order of the paper in these
proceedings in some sort of logical fashion, i.e. by subject matter
rather than chronologically according to the oral presentations. The
results are probably chaotic, understandably enough, given the topic of
the school, but since we have learned from a number of the speakers that
there is actually order in chaos, it hopefully will not matter.
The organizers would like to take the opportunity to express their
appreciation to all of the individuals who assisted in the organization
of the school, especially Orazio Muscato, Daniela Auditore and Gloria
Henneke. In addition, Ms. Henneke made heroic contributions to the
preparation of these proceedings. Without her efforts, there would have
been no proceedings.
P. Z. Zweifel
Center for Transport Theory and Mathematical Physics
Virginia Polytechnic Institute and State University
Blacksburg, Virignia 24061
CONTENTS
INTRODUCTION
Conference Summary ............................................... 1
Giovanni Gallavotti
CHAOTIC MAPS
Complicated Objects on Regular Trees ........ ................. ...... 5
Mitchell Feigenbaum
Renormalization Group Methods for Circle Mappings ........... ....... 25
Oscar E. Lanford, III
Phase Transitions on Strange Sets .................................. 37
Predrag Cvitanovic
Stochastic Perturbations of the Invariant Measure of Some
Hyperbolic Dynamical Systems ............... ......... ..... .... 55
Pierre Collet
Fixed Points of Composition Operators ....... ..... ......... ...... ... 71
Henri Epstein
Differentiable Structures on Fractal Like Sets,. ... ......... ..... ... 101
Determined by Intrinsic Scaling Functions on Dual Cantor Sets
Dennis Sullivan
CLASSICAL MECHANICS-PARTICLES AND FLUIDS
Painleve Property and Integrability ................................ 111
Eric Siggia
Nekhoroshev-like Results for Hamiltonian Dynamical Systems ......... 121
Giancarlo Bennetin
Relaxation Times and the Foundations of Classical Statistical
Mechanics in the Light of Modern Perturbation Theory. ........ 147
Luigi Galgani
Relevance of Exponentially Large Time Scales in Practical
Applications: Effective Fractal Dimensions in Conservative
Dynamical Systems ............................................ 161
Antonio Giorgilli
vii
Numerical Results from Truncated Navier-Stokes Equations
(Summary) .................................................... 171
V. Franceschini
A Simple and Compact Presentation of Birkhoff Series ............... 173
Michel Vittot
Two Lectures on Chaotic Dynamics in the Solar System ............... 185
Jack Wisdom
QUANTUM CHAOS
Quantum Chaology of Energy Levels; Notes Based on Lectures by
Michael Berry ............." ................................... 189
Jonathan Keating and Raul Mondragon
Quantum Mechanics and Chaos ......................... " . . . . . . . . . . . . .. 197
Giulio Casati
STATISTICAL MECHANICS
Dynamics of Automota, Spin Glasses and Neural Network Models 213
B. Derrida
On the Weakly Coupled Lorentz Gas .................................. 245
Detlef IXirr
Dynamical System Describing the Low Temperature Phase of a Driven
Lattice Gas .................................................. 255
Joachim Krug and Herbert Spohn
Computing Bounds on Critical Indices 269
Hans Koch and Peter Wittwer
A Thermodynamic Active System in Regular and Chaotic Motion ........ 279
R. Gallimbeni, M. Miari and L. Sertorio
EXPERIMENTS
Experiment and Theory of the Classically Chaotic Motion of the
Bound Electron ............................................... 289
James E. Bayfield
Shil'nikov Chaos in Lasers 301
F. T. Arecchi
Recent Results of Experiments with Staffman-Taylor Flow............ 313
Mark W. DiFrancesco
Instabilities and Chaos in Rotating Fluids ......................... 319
Harry L. Swinney
From Chaos to Turbulence in an Helium Experiment (Summary) ......... 327
Albert Libchaber
viii
Complex Dynamics in Experiments on a Chemical Reaction (Summary) ... 329
W. D. McCormick
ADDENDA
Additional Contributed Seminars 331
Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Index..... .. . . ... .... . ... .. . .. .. . .. . .. . .. . ... .. . .. ... . .. . . . . . . . . . . . . 335
ix
CONFERENCESUNrndARY
Giovanni Gallavotti
Centro Linceo Interdisciplinare 'B. Segre'
Accademia dei Lincei
Via della Lungara. 10
00100 Roma, Italla
This is a rather faithful transcript of the concluding remarks: no real editing has been
attempted, the purpose being to try to convey the flavour the comments made had at the end
of the school.
The uniform circular motion of Greek science dominated the scene until Poincare. Newton
produced firm a priori basis for predicting its frequencies in the most fundamental mechanical
systems like the planets, rigid bodies, oscillators, waves, etc.
Perturbation theory was greatly developed to understand the quasi-periodic structure of
more complex systems; however its growth led to the discoveries, by Poincare, of the existence
of motions fundamentally different from the quasi-periodic ones.
The work of Poinc:m~ was properly understood only by few scientists and became widely
appreciated only only after the work of Kolmogorov, Arnold, Moser, of Fermi, Pasta, Ulam and of
Lorenz, Ruelle, Takens. We have witnessed in the last twenty years the diffusing awareness of the
existence of substantially complicated motions and growing efforts towards their classification.
The conceptual necessity of the existence and relevance of non quasi-periodic motions, put
forward by Ruelle led, directly or indirectly, to many discoveries. Most of them have been at
least mentioned in this school.
The period doubling phenomenon and the various scaling structures associated with it have
been studied in detail beginning with Feigenbaum who has presented their illuminating connec
tion with the Statistical Mechanics of one dimensional lattice systems.
Among the many related topics, we had the opportunity of hearing Sullivan's characteriza
tion of the stable manifold of the Feigenbaum fixed point and Epstein's theory of the existence
of the fixed point. Collet presented us a discussion on the effects of noise on the bifurcations
heralding the chaotic transition under the period-doubling scenario, discussing also the related
problems of the effects of noise on the motion on a strange attractor. We saw that, although a
chaotic system can also be defined as a system which does not become more chaotic if noise is