Table Of ContentSpringer Series in Synergetics
Synergetics, an interdisciplinary field of research, is concerned with the cooper
ation ofi ndividual parts of a system that produces macroscopic spatial, temporal
or functional structures. It deals with deterministic as well as stochastic processes.
Volume I 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. Giittinger and H. Eikemeier
Structural Stability
in Physics
Proceedings of Two International Symposia on
Applications of Catastrophe Theory and
Topological Concepts in Physics
Tlibingen, Fed. Rep. of Germany,
May 2-6 and December 11-14, 1978
Editors: W Giittinger and H. Eikemeier
With 108 Figures
Springer-Verlag Berlin Heidelberg New York 1979
Professor Dr. Werner Gilttinger
Institute for Information Sciences, University ofTiihingen
D-7400 Tiihingen, Fed. Rep. of Germany, and
Department of Physics and Astronomy, University ofW yoming,
Laramie, WY 82071, USA
Dr. Horst Eikemeier
Institute for Information Sciences, University ofTiibingen
D-7400 Tiihingen, Fed. Rep. of Germany
ISBN-13:978-3-642-67365-8 e-ISBN-13:978-3-642-67363-4
DOl: 10.1007/978-3-642-67363-4
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© by Springer-Verlag Berlin Heidelberg 1979
Softcover reprint of the hardcover 1st edition 1979
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fore free for general use.
2153/3130-543210
Preface
This volume is the record and product of two International Symposia on the Appli
cation of Catastrophe Theory and Topological Concepts in Physics, held in May and
December 1978 at the Institute for Information Sciences, University of TUbingen.
The May Symposium centered around the conferral of an honorary doctorate upon
Professor Rene Thom, Paris, by the Faculty of Physics of the University of TUbingen
in recognition of his discovery of universal structure principles and the new di
mension he has added to scientific knowledge by his pioneering work on structural
stability and morphogenesis. Owing to the broad scope and rapid development of the
field, the May Sympos,ium was followed in December by a second one on the same sub
jects. The symposia, attended by more than 50 scientists, brought together mathe
maticians, physicists, chemists and biologists to exchange ideas about the recent
faSCinating impact of topological concepts on the physical sciences, and also to
introduce young scientists to the field. The contributions, covering a wide spectrum,
are summarized in the subsequent Introduction.
The primary support of the Symposia was provided by the "Vereinigung der Freunde
der Univertat TUbingen" (Association of the Benefactors of the University). We are
particularly indebted to Dr. H. Doerner for his personal engagement and efficient
help with the projects, both in his capacity as Secretary of the Association and as
Administrative Director of the University. We also would like to thank the President
of the University, A. Theis, for his interest and encouragement. The December Sym
posium received additional support from the British Council and the German Academic
Exchange Service. We are deeply indebted to their respective officers, Miss L.J.
Bennet and Dr. G. Schulz, for their kind cooperation.
It is a most pleasant task to express on behalf of the lecturers and participants
our gratitude to Miss E. Kohn and Miss B. Santorini for their invaluable and ef
ficient assistance in the organization of the Symposia and for their expert and
cheerful help in preparing difficult manuscripts. Furthermore, we are grateful to
Professor M. Dal Cin, and, in particular, to Dr. E. Dilger for their help and also
to Dr. H. Lotsch of Springer-Verlag for his patient cooperation.
TUbingen, March 1979 w. Gattinger
H. Eikemeier
v
Contents
Introduction ............................................. 1
~..................
Part I Gene1.'a.l. Concepts
Towards a Revival of Natural Philosophy. By R. Thorn •.••••••••••.•••••••••••• 5
Catastrophe Theory. By E.C. Zeeman (With 7 Figures) ••••••.•••••••••••••••••. 12
Catastrophe Geometry in Physics: A Perspective. By W. GUttinger ••••.••••.••• 23
Synergetics and a New Approach to Bifurcation Theory. By H. Haken •••.•.••••• 31
Part II Topol.ogical. Aspects of Wave Motion
Catastrophe and Fractal Regimes in Random Waves. By M.V. Berry
(With 2 Figures) ......................................................... 43
Distribution of Modes in Fractal Resonators. By M.V. Berry •.•••••••••..••••• 51
Optical Caustics and Diffraction Catastrophes. By J.F. Nye
(With 7 Figures) ••••••••.•.•••••••••.••••••••••.•••••.•.•..•••••••••••••• 54
Part I I I Catastrophes in Infinite Dimensions
Infinite Dimensions and the Fold Catastrophe. By R. Magnus and T. Poston
(With 14 Figures) ........................................................ 63
Catastrophes and Bifurcations in Variational Problems. By G. Dangelmayr
(With 4 Figures) •.•••••••••.•.••••••.•••••••••.••••.•••••••••••••••••.••• 84
Semiclassical Path Integrals in Terms of Catastrophes
By G. Dangelmayr, W. GUttinger, and W. Veit (With 6 Figures) 104
Catastrophe and Stochasticity in Semiclassical Quantum Mechanics
By M.V. Berry (With 1 Figure) ............................................ 122
Part IV Defects and Disz.ocations
Stable Defects in 6rdered Media. By R. Thorn (With 1 Figure) 129
Structural Stability in Evolving Flow Fields. By J.F. Nye (With 3 Figures) 134
Wavefront Dislocations and Their Analysis Using Catastrophe Theory
By F.J. Wright (With 10 Figures) ......................................... 141
VII
Part V Statistical Mechanics· and Phase Transitions
Structural Stability in Statistical Mechanics. By Rasetti ................ 159
t~.
The 1800 Rule at Triple Points: A Consideration Based on a Modified Butterfly
Model. By G. Dukek (With 5 Figures) ...................................... 176
Phase Diagrams and Catastrophes. By K. Keller. G. Dangelmayr. and H. Eikemeier
(Wi th 9 Figures) . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 186
Information Measures and Thermodynamic Criteria of Motion. By F. Schlogl 199
Phase Transitions as Catastrophes: A Perspective. By W.T. Grandy
(With 2 Figures) ..............................•..............•........... 210
Part VI Solitons
Solitons in Physics: Basic Concepts. By R.K. Bullough and R.K. Dodd
(With 1 Figure) ..........•............................................... 219
Solitons in Mathematics. By R.K. Bullough and R.K. Dodd •.................... 233
Spin Wave Breathers in 3He B. By P.W. Kitchenside. R.K. Bullough. and
P.J. Caudrey (With 10 Figures) ........................................... 254
Part VII Dynamical Systems
Catastrophic Effects in Pattern Recognition. By R. Brause and M. Dal Cin
(With 5 Figures) ......................................................... 269
Cusp Bifurcation in Pituitary Thyrotropin Secretion. By F.J. Seif
(Wi th 4 Fi gures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 275
Chaos. By O.E. Rossler (With 17 Figures) 290
Index of Contributors 311
VIII
Inttuduction
The Symposia on Structural Stability in Physics had the objective to discuss recent
applications of topological concepts deriving from the notion of structural stabil
ity -- notably, catastrophe and singularity theory -- in disparate physical systems
exhibiting analogous behavior on different scales. The flow of ideas of the central
lecture topics of this rapidly expanding field, which are at the frontier of cur
rent research, reflects the organization of these Proceedings:
The first part of the book, GENERAL CONCEPTS, begins with R. Thom's reflections
about the need for a revival of natural philosophy and the role catastrophe theory
plays in it. E.C. Zeeman then summarizes the use of catastrophes for modeling in
the sciences, and W. GUttinger provides a sketch of recent trends in applying cata
strophe theory to physics. H. Haken closes out the chapter by linking synergetics
with catastrophe and bifurcation theory.
The book then turns, in the chapter on TOPOLOGICAL ASPECTS OF WAVE MOTION, to
recent developments in the theory of random waves, fractals and diffraction cata
strophes. This part starts with M.V. Berry's surveys of the roles catastrophes and
fractals play in the theory of intensity fluctuations in random waves and in the
distribution of modes in fractal resonators. J.F. Nye concludes the chapter with
an analysis of stable optical caustics generated by irregular, shape-changing
water drop lenses and the emerging umbilic diffraction patterns.
The papers of the chapter on CATASTROPHES IN INFINITE DIMENSIONS deal with
attempts to use catastrophe techniques in spaces with infinitely many dimensions,
i.e., in continuum physics, and establish links with bifurcation theory. R. Magnus
and T. Poston begin by showing in functional analytic terms how catastrophe theo
retic methods can be used to gain insight into modeling infinite-dimensional
systems. G. Dangelmayr analyzes bifurcation phenomena in variational problems by
means of catastrophes. G. Dangelmayr, W. GUttinger and W. Veit then discuss semi
classical approximations of path integrals on and near caustics in terms of cata
strophe theory. M.V. Berry concludes the chapter by analyzing the bifurcation of
quantum bound states in the semiclassical limit using the notion of quantum maps.
The contributions of the chapter on DEFECTS AND DISLOCATIONS center around the
topological classification of singularities in ordered media, flow fields and wave
fronts. R. Thom begins by developing new ideas about combining the notion of pseudo
group with catastrophe theory to describe stable defects in ordered media. J.F. Nye
then discusses the changing topology of evolving flow fields, describes their struc
turally stable singularities in terms of catastrophes and applies the results to
geophysical phenomena. F.J. Wright concludes the chapter by showing how catastrophe
theory can be used to analyze and classify dislocations of wavefronts encountered
in various branches of physics.
The next chapter is devoted to topics in STATISTICAL MECHANICS AND PHASE TRAN
SITIONS. M. Rasetti begins with a thorough discussion of the fundamental role played
by the notion of structural stability in statistical mechanics. G. Dukek then ana
lyzes the validity of the 1800 rule of phase diagrams of thermodynamics in terms
of a butterfly catastrophe model. K. Keller, G. Dangelmayr and H. Eikemeier show
how to describe qualitatively and quantitatively various critical phenomena in
terms of higher order catastrophes. F. Schlogl then discusses generalized stabil
ity criteria for nonequilibrium phase transitions. W.T. Grandy concludes the chap
ter by developing ideas about how catastrophe theory may focus, via the structure
of partition functions, on the ultimate physical origin of phase transition pheno
mena.
The papers of the chapter on SOLITONS deal with both the mathematical aspects
and the physical significance of the soliton concept, whose topological background
is still almost unexplored. R.K. Bullough and R.K. Dodd begin with a comprehensive
survey of the role solitons play in various branches of physics and then move to
their mathematical description. P.W. Kitchenside, R.K. Bullough and P.J. Caudrey
close out the chapter with a discussion of spin wave breathers in 3He B in terms
of the double sine-Gordon equation.
The chapter on DYNAMICAL SYSTEMS crosses the borders of physics. R. Brause and
M. Dal Cin report on applications of catastrophe theory to discontinuous phenomena
in pattern recognition, learning algorithms and decision making. F.J. Seif then
models experimental results of the complex system of pituitary thyrotropin secre
tion in terms of catastrophes at the macroscopic and microscopic level. O.E. Rossler
concludes the chapter with an overall view on chaos with its many fascinating pro
blems.
In summing up, the papers of these Proceedings des·cribe,topological textures of
phYSics as well as diverse points of view towards the phYSical forces inducing to
pologies. This reflects the continuing objective of the TUbingen symposia: encourage
ment of new approaches to the deep interplay between mathematics and physics.
The Editor'S
2
Part I
General Concepts
