Table Of ContentSpringer Series in Synergetics
Editor: Hermann Haken
Synergetics, an interdisciplinary field of research, is concerned with the cooper
ation ofindividual parts of a system that produces macroscopic spatial, temporal
or functional structures. It deals with deterministic as well as stochastic processes.
Volume 1 Synergetics An Introduction 2nd Edition
ByH. Haken
Volume 2 Synergetics A Workshop
Editor: H. Haken
Volume 3 Synergetics Far from Equilibrium
Editors: A. Pacault and C. Vidal
Volume 4 Structural Stability in Physics
Editors: W. Glittinger and H. Eikemeier
Volume 5 Pattern Fonnation by Dynamic Systems and
Pattern Recognition
Editor: H. Haken
Volume 6 Dynamics of Synergetic Systems
Editor: H. Haken
Volume 7 Problems of Biological Physics
By L. A. Blumenfeld
Volume 8 Stochastic Nonlinear Systems
in Physics, Chemistry, and Biology
Editors: L. Arnold and R. Lefever
Volume 9 Numerical Methods in the Study of
Critical Phenomena
Editors: J. Della Dora, J. Demongeot, and
B. Lacolle
Volume 10 The Kinetic Theory of Electromagnetic Processes
By Y. L. Klimontovich
Volume 11 Chaos and Order in Nature
Editor: H. Haken
Chaos and Order
in Nature
Proceedings of the International Symposium
on Synergetics at Schloll Elmau, Bavaria,
April 27 - May 2, 1981
Editor: H. Haken
With 134 Figures
Springer-Verlag Berlin Heidelberg New York 1981
Professor Dr. Hermann Haken
Institut flir Theoretische Physik der Universitat Stuttgart
Pfaffenwaldring 57/rv, D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN-13 :978-3-642-68306-0 e-ISBN-13 :978-3-642-68304-6
DOl: 10.1007/978-3-642-68304-6
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© by Springer-Verlag Berlin Heidelberg 1981
Softcover reprint of the hardcover 1st edition 1981
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2153/3130-543210
Preface
This book contains the invited papers of an international symposium on synergetics;
which was held at Schlol3 Elmau, Bavaria, FRG, April 27 to May 1, 1981. At our
previous meetings on synergetics the self-organized formation of structures in
quite different disciplines stood in the foreground of our interest. More recently
it has turned out that phenomena characterized by the word "chaos" appear in various
disciplines, and again far-reaching analogies in the behavior of quite different
systems become visible. Therefore this meeting was devoted not only to problems
connected with the occurrence of ordered structures but also to most recent results
obtained in the study of chaotic motion. In the strict mathematical sense we are
dealing here with deterministic chaos, i.e., irregular motion described by deter
ministic equations. While in this relatively young fieJd of research computer ex
periments and computer simulations predominated in the past, there now seems to be
a change of trend, namely to study certain regular features of chaos by analytical
metbods. I think considerable progress has been achieved in this respect quite
recently. This theoretical work is paralleled by a number of very beautiful experi
ments in different fields, e.g., fluid dynamics, solid-state physics, and chemistry.
For the first time at this kind of meeting we have included plasma physics,
which presents a number of most fascinating problems with respect to instabilities,
formation of structures, and related phenomena.
I hope that the readers will find this volume as useful as our previous ones,
giving a survey on experimental and theoretical studies on synergetic systems.
This symposium was made possible by a grant from the Volkswagenwerk Foundation,
Hannover, and I would like to take this opportunity to thank the foundation for
its continuous and very effective support of synergetics.
lowe particular thanks to my secretary, Mrs. U.Funke, for her great and indis
pensable help in organizing this symposium and for her assistance in editing these
proceedings.
Stuttgart, September 1981 H. Haken
v
Contents
Part I Introduction
Chaos and Order in Nature
By H. Haken (With 7 Figures) 2
Part II Fluid Dynamics. Order and Chaos in Fluid Dynamics
Rayleigh-Benard Convection in High Prandtl Number Fluid
By P. Berge (With 11 Figures) ......................................... 14
Rayleigh-Benard Experiment in a Low Prandtl Number Fluid, Mercury
By S. Fauve, A. Libchaber (With 8 Figures) ............................ 25
Transition to Turbulence Via the Statistical Limit Cycle Route
By F. H. Busse (With 6 Figures) ........................................ 36
Divergence of Coherence Length and Excitation of Resonance in Taylor Vortex
Flow. By A. Brandstater, G. Pfister, I. Rehberg, E.O. Schulz-DuBois
(With 5 Figures) ...................................................... 45
Non-Equilibrium Phase Transitions in a Kundt's Tube
By A. HUbler, G. Schubert, G. Meyer-Kress (With 3 Figures) 49
Part III Chaos in Fluids, Solid State Physics, and Chemical Reactions
Stochastization of Coherent Structures by a Periodic Field
By I.S. Aranson, M.I. Rabinovich, M.M. Sushchik (With 6 Figures) 54
Turbulence and Scaling in Solid State Physics
By B.A. Huberman ...................................................... 64
Dynamic Instabilities Observed in the Belousov-Zhabotinsky System
By C. Vidal (With 6 Figures) .......................................... 69
Part IV Instabilities and Bifurcations: Theoretical Approaches
Hopf-Landau Bifurcation Near Strange Attractors
By G. R. Sell . ... . ... ... ... . .. . .. . .. . ... . .. . . ... .. .. .... .. . . . .... .... .. 84
Dispersive Instabilities in Nonlinear Systems: The Real and Complex Lorenz
Equation. By J.D. Gibbon (With 3 Figures) ............................. 92
Bifurcations and Multistability in Nonlinear Optics
By D.F. Walls, P. Zoller, P.O. Drummond, C.V. Kunasz (With 8 Figures) 102
VII
Part V Plasma Instabilities
Coherent Wave Interactions in Plasmas and Active Molecular Media
By H. Wil helmsson (With 1 Figure) ..................................... 112
Instability as a Property of Plasma States
By E. RKuchle (With 12 Figures) ....................................... 118
Phenomena of Self Organization in Dense Plasma
By H. Krompholz, G. Herziger (With 10 Figures) 131
Part V I Phase Transitions
Closed-Form Approximation and Interpolation Formulae for the 3-Dimensional
Ising Model. By Bai-lin Hao (With 3 Figures) .... ... ................... 144
Part VII Path Integrals: Recent Developments
Path Integral Approach to Fluctuations in Dynamic Processes
By H. Leschke ......................................................... 156
Definitions of Path Integrals for General Diffusion Processes
By Ch. Wi sse I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
The Uses of Path Integrals for Diff~sion in Bistable Potentials
By U. Weiss (With 6 Figures) .......................................... 177
Part V III General Systems Approaches
Structural Stability of Stochastic Systems
ByW. Ebeling ......................................................... 188
Structure-Building Phenomena in Systems with Power-Product Forces
By W. Mende, M. Peschel (With 1 Figure) ... ................... .... ..... 196
Part IX Morphogenesis
Spatial-Temporal Coordination of Mitotic Activity in Developing Sea Urchin
Embryos. By E. Parisi, S. Filosa, A. Monroy (With 6 Figures) .......... 208
Part X Once Again in Chaos: Theoretical Approaches
Modeling Chaotic Systems
By R. Shaw (With 7 Figures) 218
Feigenbaum Sequences in Conservative and Dissipative Systems
By R.H.G. Helleman (With 6 Figures) ................................... 232
On the Perfect Delay Convention or the Revolt of the Slaved Variables
By M. Diener, T. Poston (With 23 Figures) .......... ................ ... 249
The Mechanism by Which Many Partial Difference Equations Destabilize
By W. Briggs, A.C. Newell, T. Sarie (With 5 Figures) .................. 269
List of Contributors 275
VIII
Part I
Introduction
Chaos and Order in Nature
Hermann Haken
Institut fUr Theoretische Physik der Universitat Stuttgart
0-7000 Stuttgart 80, Fed. Rep of Germany
1. Introduction
Interdisciplinary meetings are nowadays quite fashionable, but bring
ing together scientists from different disciplines does not necessarily
guarantee the success of such a meeting, for instance, in the sense
that a new coherent idea emerges. Therefore a few words might be in
order for those readers who are not familiar with the aims of syner
getics or a meeting on this topic.
Synergetics deals with systems composed of many subsystems. The
main aims of synergetics can be well visualized by means of an example
taken from biology. Figure 1 shows two different kinds of fish . But as
was found by O'A.Thompson at the beginning of this century, the shapes
of the two fishes can be transformed into each other by a rather
simple grid transformation. This is indeed quite a remarkable finding
Fig.1 Two different kinds of fish (porcu
pine fish Oiodon, left, and sunfish Ortha
goriscus mola, right) can be "transformed"
into each other (after O'Arcy Wentworth
Thompson, "On Growth and Form", 1917).
Compare text.
and represents one of the methods nature applies to produce new
forms. On the other hand the study of such phenomena is not the main
goal of synergetics. Indeed a mathematician would perhaps not consider
these two kinds of fish as different. He rather would say that we have
here a case of structural stabil ity in front of us. We have a mere
deformation but not a structural change, for instance the occurrence
of a new additional fin. On the other hand biology provides us with a
wealth of examples in which structural changes occur, for instance
in embryology. An example is given in fig. 2. Here in the development
of the California newt dramatic qualitative changes occur in succession,
for instance incisions, and later on the appearance of extremities
which in a way can be considered as singularities of otherwise smooth
surfaces.
Here we are now in the middle of the problems of synergetics, name
ly the study of those phenomena where patterns change qualitatively
and dramatically on macroscopic scales. Such changes of patterns may
refer to space, time, or processes in space and time. Over the past,
2
Description:This book contains the invited papers of an international symposium on synergetics; which was held at Schlol3 Elmau, Bavaria, FRG, April 27 to May 1, 1981. At our previous meetings on synergetics the self-organized formation of structures in quite different disciplines stood in the foreground of our
