Table Of ContentSpringer Series in Synergetics
Editor: Hermann Haken
Synergetics, an interdisciplinary field of research, is concerned with the cooper
ation of individual parts of a system that produces macroscopic spatial, temporal
or functional structures. It deals with deterministic as well as stochastic processes.
1 Synergetics An Introduction 3rd Edition 21 Stochastic Phenomena and Chaotic Behaviour
ByH. Haken in Complex Systems Editor: P. Schuster
2 Synergetics A Workshop 22 Synergetics - From Microscopic to Macroscopic
Editor: H. Haken Order Editor: E. Frehland
3 Synergetics Far from Equilibrium 23 Synergetics of the Brain
Editors: A. Pacault, C. Vidal Editors: E. Ba§ar, H. Flohr, H. Haken,
4 Structural Stability in Physics A.J. Mandell
Editors: W. Giittinger, H. Eikemeier 24 Chaos and Statistical Methods
5 Pattern Formation by Dynamic Systems and Editor: Y. Kuramoto
Pattern Recognition 25 Dynamics of merarchical Systems
Editor: H. Haken An Evolutionary Approach By J.S. Nicolis
6 Dynamics of Synergetic Systems 26 Self-Organization and Management of
Editor: H. Haken Social Systems
7 Problems of Biological Physics Editors: H. Ulrich, G.J.B. Probst
By L. A. Blumenfeld 27 Non-Equilibrium Dynamics in Chemical
8 Stochastic Nonlinear Systems Systems Editors: C. Vidal, A. Pacault
in Physics, Chemistry, and Biology 28 Self-Organization Autowaves and Structures
Editors: L. Arnold, R. Lefever Far from Equilibrium
9 Numerical Methods in the Study of Critical Editor: V.1. Krinsky
Phenomena 29 Temporal Order
Editors: J. Della Dora, J. Demongeot, Editors: L. Rensing, N.I.Jaeger
B. Lacolle 30 Dynamical Problems in Soliton Systems
10 The Kinetic Theory of Electromagnetic Editor: S. Takeno
Processes By Yu. L. Klimontovich 31 Complex Systems - Operational Approaches
in Neurobiology, Physics, and Computers
11 Chaos and Order in Nature
Editor: H. Haken
Editor: H. Haken
32 Dimensions and Entropies in Chaotic Systems
12 Nonlinear Phenomena in Chemical Dynamics
Quantification of Complex Behavior
Editors: C. Vidal, A. Pacault
Editor: G. Mayer-Kress
13 Handbook of Stochastic Methods
33 Selforganization by Nonlinear Irreversible
for Physics, Chemistry, and the Natural Sciences
Processes
2nd Edition By C. W. Gardiner
Editors: W. Ebeling, H. Ulbricht
14 Concepts and Models of a Quantitative
34 Instabilities and Chaos in Quantum Optics
Sociology The Dynamics of Interacting
Editors: F. T. Arecchi, R. G. Harrison
Populations By W. Weidlich, G. Haag
35 Nonequilibrium Phase Transitions in
15 Noise-Induced Transitions Theory and
Semiconductors Self-Organization Induced by
Applications in Physics, Chemistry, and Biology
Generation and Recombination Processes
By W. Horsthemke, R. Lefever By E. SchOll
16 Physics of Bioenergetic Processes 36 Thmporal Disorder in Human
By L. A. Blumenfeld Oscillatory Systems
17 Evolution of Order and Chaos in Physics, Editors: L. Rensing, U. an der Heiden,
Chemistry, and Biology Editor H. Haken M.C. Mackey
18 The Fokker-Planck Equation 37 The Physics of Structure Formation
ByH. Risken Theory and Simulation
19 Chemical Oscillations, Waves, and Turbulence Editors: W. Giittinger and G. Dangelmayr
By Y. Kuramoto 38 Computational Systems - Natural and
20 Advanced Synergetics 2nd Edition Artificial
By H. Haken Editor: H. Haken
The Physics of
Structure Formation
Theory and Simulation
Proceedings of an International Symposium,
Tiibingen, Fed. Rep. of Germany,
October 27 - November 2,1986
Editors: W. Guttinger and G. Dangelmayr
With 207 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo
Professor Dr. Werner Giittinger
Dr. Gerhard Dangelmayr
Institut fUr Informationsverarbeitung, Universitat Tiibingen
D-7400 Tiibingen, Fed. Rep. of Germany
Series Editor:
Professor Dr. Dr. h. c. Hermann Haken
Institut fUr Theoretische Physik der Universitiit Stuttgart, Pfaffenwaldring 57/IV,
D-7000 Stuttgart 80, Fed. Rep. of Germany and
Center for Complex Systems, Florida Atlantic University,
Boca Raton, FL 33431, USA
ISBN-13: 978-3-642-73003-0 e-ISBN-13: 978-3-642-73001-6
DOl: 10.1007/978-3-642-73001-6
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on
microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only
permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24,
1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German
Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987
Softcover reprint of the hardcover 1st edition 1987
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant protective laws and regulations and therefore free for
general use.
2153/3150-543210
Preface
The formation and evolution of complex dynamical structures is one of the most
exciting areas of nonlinear physics. Such pattern formation problems are common
in practically all systems involving a large number of interacting components.
Here, the basic problem is to understand how competing physical forces can
shape stable geometries and to explain why nature prefers just these.
Motivation for the intensive study of pattern formation phenomena during the
past few years derives from an increasing appreciation of the remarkable diversity
of behaviour encountered in nonlinear systems and of universal features shared
by entire classes of nonlinear processes. As physics copes with ever more ambi
tious problems in pattern formation, summarizing our present state of knowledge
becomes a pressing issue. This volume presents an overview of selected topics in
this field of current interest. It deals with theoretical models of pattern formation
and with simulations that bridge the gap between theory and experiment. The
book is a product of the International Symposium on the Physics of Structure
Formation, held from October 27 through November 2, 1986, at the Institute
for Information Sciences of the University of Tiibingen. The symposium brought
together a group of distinguished scientists from various disciplines to exchange
ideas about recent advances in pattern formation in the physical sciences, and
also to introduce young scientists to the fi<;ld.
Our sincere thanks are due to many. First and foremost to the lecturers
for their stimulating presentations and the extra effort they took to make them
into authoritative reports. The generous financial support for the symposium
was provided by the Stiftung Volkswagenwerk of the Federal Republic of Ger
many. We are particularly indebted to Dr. H. Plate of the Foundation for his
interest and help with our projects. Furthermore, we are pleased to thank the
Vice-President of the University, Dr. E. Lindner, and the Chancellor, Dr. G.
Sandberger, for their efficient cooperation.
It is a most pleasant task to express our gratitude to Mrs. E. Allmendinger, G.
Ambros, C. Ott, and V. Miiller for their invaluable assistance in the organization
of the symposium and to Mrs. A. Kluwe, M. Resch, and C. Trost for their help
in preparing the manuscripts. Finally, we thank our colleagues D. Armbruster,
C. Geiger, P. Haug, D. Lang, and M. Neveling for their help throughout and also
Dr. H. Lotsch of Springer-Verlag for his patient cooperation.
Tiibingen W. Giittinger
June, 1987 G. Dangelmayr
v
Contents
Introduction XI
Part I Disordered Systems, Networks and Patterns
Neural Networks - Achievements, Prospects, Difficulties
By D.J. Amit (With 9 Figures) ........................... 2
Pattern Recognition in Nonlinear Neural Networks
By J .L. van Hemmen (With 5 Figures) .. . . . . . . . . . . . . . . . . . . . . 22
Formation of Ordered Structures in Quenching Experiments: Scaling
Theory and Simulations
By D.W. Heermann and K. Binder (With 6 Figures) ........... 42
Patterns in Random Boolean Nets
By G. Weisbuch (With 14 Figures) 52
The Structure of the Visual Field
By J.J. Koenderink (With 6 Figures) 68
Patterns of Maximal Entropy
By R.D. Levine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Part II Structure Formation in Nonlinear Systems
Patterns of Bifurcation from Spherically Symmetric States
By F.H. Busse (With 2 Figures) .......................... 88
Structure Formation by Propagating Fronts
By M. Lucke, M. Mihelcic, B. Kowalski, and K. Wingerath
(With 12 Figures) ..................................... 97
Oscillatory Doubly Diffusive Convection: Theory and Experiment
By E. Knobloch, A.E. Deane, and J. Toomre (With 7 Figures) .... 117
Pattern Selection in Anisotropic Systems
By L. Kramer, E. Bodenschatz, W. Pesch, and W. Zimmermann
(With 1 Figure) ...................................... 130
VII
Rayleigh Number Ramps Cause Moving Convection Patterns
By 1. Rehberg and H. Riecke (With 10 Figures) ............... 142
Interactions of Stationary Modes in Systems with Two and Three
Spatial Degrees of Freedom
By M. Neveling, D. Lang, P. Haug, W. Giittinger, and G. Dangelmayr
(With 10 Figures) ..................................... 153
The Basic (n,2n)-Fold of Steady Axisymmetric Taylor Vortex Flows
By R. Meyer-Spasche and M. Wagner (With 9 Figures) .. . . . . . . .. 166
Part III Interfacial Patterns
Nonlinear Interactions of Interface Structures of Differing Wavelength
in Directional Solidification
By M.J. Bennett, R.A. Brown, and L.H. Ungar (With 8 Figures) 180
Velocity Selection in Dendritic Growth
By B. Caroli, C. Caroli, and B. Roulet . . . . . . . . . . . . . . . . . . . . .. 191
Interfacial Pattern Formation: Dynamical Effects
By D.A. Kessler and H. Levine (With 6 Figures) 198
Part IV Diffusion Limited Aggregation and Fractals
Role of Fluctuations in Fluid Mechanics and Dendritic Solidification
By H.E. Stanley (With 13 Figures) ........................ 210
A Stochastic Model for Arborescent and Dendritic Growth
By J. Nittmann (With 8 Figures) ......................... 245
Electrodeposition: Pattern Formation and Fractal Growth
By L.M. Sander (With 3 Figures) ......................... 257
Anisotropy and Fluctuation in Diffusion Limited Aggregation
By M. Kolb (With 5 Figures) ............................ 267
The Classes of Fractals
By B. Rohricht, \V. Metzler, J. Parisi, J. Peinke, W. Beau,
and O.E. Rossler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 275
Structural Characterization of Aluminum Hydroxide Aggregates by
S.A.X.S.: Comparison Between Simulations of Fractal Structures and
Experiments
By M.A.V. Axelos, D. Tchoubar, and R. Jullien (With 5 Figures) 282
Part V Chaos and Turbulence
Spatial Disorder in Extended Systems
By P. Coullet, C. Elphick, and D. Repaux (With 3 Figures) 290
VIII
Nonlinear Gravity Waves and Resonant Forcing
By K. Kirchgassner (With 3 Figures) ....................... 303
Transitions to Chaos in a Finite Macroscopic System: Direct
Numerical Simulations Versus Normal Form Predictions
By A. Arneodo and O. Thual (With 16 Figures) ............... 313
Time-Dependent and Chaotic Behaviour in Systems with 0(3)-
Symmetry
By R. Friedrich and H. Haken (With 10 Figures) .............. 334
Selfsimilarity of Developed Turbulence
By H. Effinger and S. Grossmann (With 2 Figures) 346
Part VI Bifurcations and Symmetry
Nonlinear Normal Modes of Symmetric Hamiltonian Systems
By J. Montaldi, M. Roberts, and 1. Stewart (With 8 Figures) 354
The Ubiquitous Astroid
By D. Chillingworth (With 19 Figures) 372
On the Hopf Bifurcation with Broken 0(2) Symmetry
By G. Dangelmayr and E. Knobloch (With 2 Figures) 387
Generic Spontaneous Symmetry Brealcing in SU (n) - Equivariant
Bifurcation Problems
By C. Geiger and W. Giittinger (With 1 Figure) .............. 394
Computer Algebraic Tools for Applications of Catastrophe Theory
By F.J. Wright and R.G. Cowell (With 4 Figures) ............. 402
Computer Algebra Programs for Dynamical Systems Theory
By D. Armbruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 416
Index of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . .. 427
IX
Introduction
The universe we see is a ceaseless creation, evolution and annihilation of struc
tures or patterns which are endowed with a degree of stability: They take up
some part of space and last for some period of time. It is a central problem
of the natural sciences to explain this change of structure and, if possible, to
predict it.
Nature surprises us with the fact that the tremendous amount of physical
data and results can be condensed into a few simple laws and equations that
summarize our knowledge. These laws are essentially qualitative and the ques
tion arises how nature generates the enormous variety and novelty of structures
by starting from such simple principles. Unaware of the scope of simple equa
tions, man has often concluded that more than mere equations is required to
explain the complexities of the world. Today, we are not so sure, for already a
simple random walk described by the Laplace equation provides a prototypical
situation with behaviours ranging from orderly to chaotic. The formation of
spatio-temporal patterns is, first of all, a geometrical problem. Thus, in our
quest to understand how structures are generated by nature, the relationship
between physical forces and the geometries they can shape is of fundamental
importance. This relationship and the ensuing universal character of nonlinear
pattern-forming processes are the central themes underlying this boole
The mechanisms that lead to pattern formation are best understood in
continuous nonlinear systems, which to external probes appear to be governed
by a few essential degrees of freedom. In such systems, structural changes are
characterized by the appearance of singularities and bifurcations which occur
if a balance existing between competing system-immanent effects breaks down.
As a consequence, an initially quiescent system becomes unstable and, in a
sequence of bifurcations, restabilizes successively in ever more complex space
and time-dependent configurations. Within particular symmetry constraints,
the bifurcations occur in certain definite and classifiable ways if the system
is structurally stable. For large continuous systems, in which whole bands of
modes become unstable, such a geometrical forrqalism is presently under de
velopment. New types of pattern formation problems occur in complex discrete
systems composed of a very large but finite number of interacting elementary
units, exemplified by neural networks and cellular automata. They possess self
organizing properties and the capacity to store and retrieve information. The
following parts of this volume present reports on these classes of systems and
the structure formation processes displayed by them on various scales.
XI
The contributions of the first part deal with DISORDERED SYSTEMS
and NETWORKS. Their scientific interest derives from the desire to design and
anlyze systems which, by means of sensory and perceptual mechanisms, can
construct internal representations of physical structures in the outside world.
The typical problems one encounters here include dynamical computing, learn
ing of patterns and adaptation. D.J. Amit begins with a survey of models of
neural networks, analyzes their pattern-forming and retrieving capabilities and
discusses future prospects in this field. J.L. van Hemmen examines the statis
tical mechanics and bifurcation properties of nonlinear neural networks. D.W.
Heermann and K. Binder then discuss the formation of ordered structures in
quenching experiments. G. Weisbuch shows how random Boolean nets develop
patterns ranging from orderly to chaotic. J.J. Koenderink guides us through
the structure of the visual field and R.D. Levine concludes the part with a
discussion of the role of the maximum entropy formalism in pattern formation.
The book then turns, in the part on STRUCTURE FORMATION in NON
LINEAR SYSTEMS, to pattern formation problems in continuum physics, with
mode interaction and symmetry constraints, which are amenable to a bifur
cation analysis. F.H. Busse begins with a discussion of pattern formation in
systems with spherical symmetry. M. Lucke, M. Mihelcic, B. Kowalski and
K. Wingerath then investigate the propagation of patterns into an unstable
homogeneous region. E. Knobloch, A.E. Deane and J. Toomre show how trav
elling, standing and modulated waves interact in doubly diffusive convection
and compare experimental results with theoretical predictions. L. Kramer, E.
Bodenschatz, W. Pesch and W. Zimmermann then discuss pattern selection
and defects in anisotropic systems related to electrohydrodynamic convection,
while J. Rehberg and H. Riecke investigate experimentally and theoretically
the induction of moving convection patterns in a cell with a ramp. M. Nevel
ing, D. Lang, P. Haug, W. Guttinger and G. Dangelmayr develop a theory of
interacting modes in systems with two and three spatial dimensions and apply
it to directional solidification and thermohaline convection. R. Meyer-Spasche
and M. Wagner close the section with an analysis of mode interaction in Taylor
vortex flows.
The papers in the part on INTERFACIAL PATTERNS survey mechanisms
by which patterns are generated by the motion of an interface between different
phases, a subject that is also of technological interest. M.J. Bennet, R.A. Brown
and L.H. Ungar begin with a discussion of the formation of cellular patterns
and their interaction in directional solidification. B. Caroli, C. Caroli and B.
Roulet then survey recent developments in our understanding of the problem
of velocity selection in dendritic growth processes. D.A. Kessler and H. Levine
conclude the part by reviewing the dynamics of side-branching and tip-splitting
in solidifying systems and in viscous flows.
The next part, on DIFFUSION LIMITED AGGREGATION and FRAC
TALS, is devoted to the study of the behaviour of aggregating particles and
structures formed by random growth. This part starts with H.E. Stanley's sur
vey of the principles of diffusion limited aggregation (DLA) and of its role in
fluid dynamics and in dendritic solidification. J. Nittmann explores the impli-
XII
