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 weH as stochastic processes.
Volume 1 Synergetics An Introduction 3rd 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. Güttinger and H. Eikemeier
Volume 5 Pattern Formation by Dynamic Systems and Pattern Recognition
Editor: H. Haken
Volume 6 Dynamies of Synergetic Systems Editor: H. Haken
Volume 7 Problems of BiologicaI 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 CriticaI Phenomena
Editors: J. Della Dora, J. Demongeot, and B. Lacolle
Volume 10 The Kinetic Theory of Electromagnetic Processes By Yu. L. Klimontovich
Volume 11 Chaos and Order in Nature Editor: H. Haken
Volume 12 Nonlinear Phenomena in ChemicaI Dynamics Editors: C. Vidal and A. Pacault
Volume 13 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences
By C. W. Gardiner
Volume 14 Concepts and Models of a Quantitative Sociology
The Dynamics of Interacting Populations By W. Weidlich and G. Haag
Volume 15 Noise-Induced Transitions Theory and Applications in Physics, Chemistry,
and Biology By W. Horsthemke and R Lefever
Volume 16 Physics of Bioenergetic Processes By L. A. Blumenfeld
Volume 17 Evolution of Order and Chaos in Physics, Chemistry, and Biology
Editor: H. Haken
Volume 18 The Fokker-Planck-Equation By H. Risken
Volume 19 ChemicaI Oscillations, Waves, and Turbulence By Y Kuramoto
Volume 20 Advanced Synergetics By H. Haken
Hermann Haken
Advanced Synergetics
Instability Hierarchies
of Self-Organizing Systems and Devices
With 105 Figures
Springer. :verlag
Berlin Heidelberg New York Tokyo 1983
Professor Dr. Dr. h.c. Herrnann Haken
Institut für Theoretische Physik der Universität Stuttgart, Pfaffenwaldring 57 IN
D-7000 Stuttgart 80, Fed. Rep. ofGermany
ISBN-13: 978-3-642-45555-1 e-ISBN-13: 978-3-642-45553-7
001: 10.1007/978-3-642-45553-7
Library of Congress Cataloging in Publication Data. Haken, H. Advanced synergetics. (Springer series in
synergetics ; v. 20) 1. System theory. 2. Self-organizing systems. L Title. 11. Series. Q295.H34 1983 003 83-340
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© by Springer-Verlag Berlin Heidelberg 1983
Softcover reprint of the hardcover 1s t ed ition 1983
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.
Typesetting: K + V Fotosatz, Beerfelden
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to Edith,
Maria, Karin, and Karl-Ludwig
Preface
This text on the interdisciplinary field of synergetics will be of interest to students
and scientists in physics, chemistry, mathematics, biology, electrical, civil and
mechanical engineering, and other fields. It continues the outline of basic con
cepts and methods presented in my book Synergetics. An Introduction, which
has by now appeared in English, Russian, J apanese, Chinese, and German.
I have written the present book in such a way that most of it can be read in
dependently of my previous book, though occasionally some knowledge of that
book might be useful.
But why do these books address such a wide audience? Why are instabilities
such a common feature, and what do devices and self-organizing systems have in
common? Self-organizing systems acquire their structures or functions without
specific interference from outside. The differentiation of cells in biology, and the
process of evolution are both examples of self-organization. Devices such as the
electronic oscillators used in radio transmitters, on the other hand, are man
made. But we often forget that in many cases devices function by means of pro
cesses which are also based on self-organization. In an electronic oscillator the
motion of electrons becomes coherent without any coherent driving force from
the outside; the device is constructed in such a way as to permit specific collective
motions of the electrons. Quite evidently the dividing line between self-organiz
ing systems and man-made devices is not at all rigid. While in devices man sets
certain boundary conditions which make self-organization of the components
possible, in biological systems aseries of self-imposed conditions permits and
directs self-organization. For these reasons it is useful to study devices and self
organizing systems, particularly those of biology, from a common point of view.
In order to elucidate fully the subtitle of the present book we must explain
what instability means. Perhaps this is best done by giving an example. A liquid
in a quiescent state which starts a macroscopic oscillation is leaving an old state
and entering a new one, and thus loses its stability. When we change certain
conditions, e. g. power input, a system may run through aseries of instabilities
leading to quite different patterns of behavior.
The central question in synergetics is whether there are general principles
which govern the self-organized formation of structures and/or functions in both
the animate and the inanimate world. When I answered this question in the af
firmative for large classes of systems more than a decade ago and suggested that
these problems be treated within the interdisciplinary research field of "synerge
tics", this might have seemed absurd to many scientists. Why should systems
consisting of components as different as electrons, atoms, molecules, photons,
cells, animals, or even humans be governed by the same principles when they or-
VIII Preface
ganize themselves to form electrical oscillations, patterns in fluids, chemical
waves, laser beams, organs, animal societies, or social groups? But the past
decade has brought an abundance of evidence indicating that this is, indeed, the
case, and many individual examples long known in the literature could be sub
sumed under the unifying concepts of synergetics. These examples range from
biological morphogenesis and certain aspects of brain function to the flutter of
airplane wings; from molecular physics to gigantic transformations of stars;
from electronic devices to the formation of public opinion; and from muscle con
traction to the buckling of solid structures. In addition, there appears to be a re
markable convergence of the basic concepts of various disciplines with regard to
the formation of spatial, temporal, and functional structures.
In view of the numerous links between synergetics and other fields, one might
assurne that synergetics employs a great number of quite different concepts. This
is not the case, however, and the situation can best be elucidated by an analogy.
The explanation of how a gasoline engine functions is quite simple, at least in
principle. But to construct the engine of a simple car, a racing car, or a modern
aircraft requires more and more technical know-how. Similarly, the basic
concepts of synergetics can be explained rather simply, but the application of
these concepts to real systems calls for considerable technical (i. e. mathematical)
know-how. This book is meant to serve two purposes: (1) It offers a simple pre
sentation of the basic principles of instability, order parameters, and slaving.
These concepts represent the "hard core" of synergetics in its present form and
enable us to cope with large classes of complex systems ranging from those of the
"hard" to those of the "soft" sciences. (2) It presents the necessary mathematical
know-how by introducing the fundamental approaches step by step, from simple
to more complicated cases. This should enable the reader to apply these methods
to concrete problems in his own field, or to search for further analogies between'
different fields.
Modern science is quite often buried under heaps of nomenclature. I have
tried to reduce it to a minimum and to explain new words whenever necessary.
Theorems and methods are presented so that they can be applied to concrete
problems; i. e. constructions rather than existence proofs are given. To use a con
cept of general systems theory, I have tried to present an operational approach.
While this proved feasible in most parts of the book, difficulties arose in connec
tion with quasi-periodic processes. I have included these intricate problems (e. g.,
the bifurcation of tori) and my approach to solving them not only because they
are at the frontier of modern mathematical research, but also because we are con
fronted with them again and again in natural and man-made systems. The
chapters which treat these problems, as weIl as some other difficuIt chapters,
have been marked with an asterisk and can be skipped in a first reading.
Because this book is appearing in the Springer Series in Synergetics, a few
words should be said about how it relates to other books in this series. Volume 1,
Synergetics. An Introduction, dealt mainly with the first instability leading to
spatial patterns or oscillations, whereas the present book is concerned with all
sorts of instabilities and their sequences. While the former book also contained
an introduction to stochastic processes, these are treated in more detail in C. W.
Gardiner's Handbook 0/ Stochastic Methods, which also provides a general
Preface IX
background for the forthcoming volumes Noise-Induced Transitions by W.
Horsthemke and R. Lefever and The Fokker-Planck Equation. Methods 01 Solu
tion and Applications by H. Risken. These three books cover important aspects
of synergetics with regard to fluctuations. The problem of multiplicative noise
and its most interesting consequences are treated only very briefly in the present
book, and the interested reader is referred to the volume by Horsthemke and
Lefever. The Fokker-Planck equation, a most valuable tool in treating systems at
transition points where other methods may fail, is discussed in both my previous
book and this one, but readers who want a comprehensive account of present
day knowledge on that subject are referred to the volume by Risken. Finally, in
order to keep the present book within a reasonable size, I refrained from going
into too far-reaching applications of the methods presented. The application of
the concepts and methods of synergetics to sociology and economy are given
thorough treatment in W. Weidlich and G. Haag's book, Concepts and Models
01 a Quantitative Sociology. Finally, Klimontovich's book, The Kinetic Theory
01 Electromagnetic Processes, gives an excellent account of the interaction of
charged particles with electromagnetic fields, as well as the various collective
phenomena arising from this interaction.
While the present book and the others just mentioned provide us with the
theoretical and mathematical basis of synergetics, experiments on self-organizing
systems are at least equally important. Thus far, these experiments have only
been treated in conference proceedings in the Springer Series in Synergetics; it is
hoped that in future they will also be covered in monographs within this series. In
conclusion, it should be pointed out that synergetics is a field which still offers
great scope for experimental and theoretical research.
I am grateful to Dipl. Phys. Karl Zeile and Dr. Arne Wunderlin for their
valuable suggestions and their detailed and careful checking of the manuscript
and the calculations. My thanks also go to Dr. Herbert Ohno, who did the draw
ings, and to my secretary, Mrs. U. Funke, who meticulously typed several ver
sions of the manuscript, including the formulas, and without whose tireless ef
forts this book would not have been possible. I gratefully acknowledge the excel
lent cooperation of the members of Springer-Verlag in producing this book. I
wish to thank the Volkswagenwerk Foundation, Hannover, for its very efficient
support of the synergetics project, within the framework of which a number of
the results presented in this book were obtained over the past four years.
Stuttgart, January 1983 Hermann Haken
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What is Synergetics About? ................................ 1
1.2 Physics ................................................. 1
1.2.1 Fluids: Formation ofDynamic Patterns..... ...... ... . . 1
1.2.2 Lasers: Coherent Oscillations ........................ 6
1.2.3 Plasmas: A Wealth of Instabilities .................... 7
1.2.4 Solid-State Physics: Multistability, Pulses, Chaos ....... 8
1.3 Engineering.............................................. 9
1.3.1 Civil, Mechanical, and Aero-Space Engineering:
Post-Buckling Patterns, Flutter, etc. .................. 9
1.3.2 Electrical Engineering and Electronics: Nonlinear
Oscillations ....................................... 9
1.4 Chemistry: Macroscopic Patterns ........................... 11
1.5 Biology ................................................. 13
1.5.1 Some General Remarks ............................. 13
1.5.2 Morphogenesis .................................... 13
1.5.3 Population Dynamies. . . .. . . . . . . . . . . . . . .. . .. . .. . . . . . 14
1.5.4 Evolution......................................... 14
1.5.5 Immune System. . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . . . . . 15
1.6 Computer Sciences ................. , .......... , . .. . .. . . . . . 15
1.6.1 Self-Organization of Computers, in Particular
Parallel Computing . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 15
1.6.2 Pattern Recognition by Machines ..................... 15
1.6.3 Reliable Systems from Unreliable Elements . . . . . . . . . . . . . 15
1.7 Economy................................................ 16
1.8 Ecology................................................. 17
1.9 Sociology ............................................... 17
1.10 What are the Common Features of the Above Examples? ....... 17
1.11 The Kind of Equations We Want to Study .................... 18
1.11.1 DifferentialEquations .............................. 19
1.11.2 First-Order Differential Equations .................... 19
1.11.3 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.11.4 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.11.5 Stochasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.11.6 Many Components and the Mezoscopic Approach. .... . . 22
1.12 How to Visualize the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.13 Qualitative Changes: General Approach. . . . . . . . . . . . . . . . . . . . . . 32
1.14 Qualitative Changes: Typical Phenomena .................... 36
XII Contents
1.14.1 Bifurcation from One Node (or Focus) into Two Nodes
(or Foci) .......................................... 37
1.14.2 Bifurcation from a Focus into a Limit Cycle
(Hopf Bifurcation) ................................. 38
1.14.3 Bifurcations from a Limit Cycle ...................... 39
1.14.4 Bifurcations from a Torus to Other Tori ............... 41
1.14.5 Chaotic Attractors ................................. 42
1.14.6 LyapunovExponents* .............................. 42
1.15 The Impact of Fluctuations (Noise). Nonequilibrium Phase
Transitions .............................................. 46
1.16 Evolution of Spatial Patterns ............................... 47
1.17 Discrete Maps. The Poincare Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.18 Discrete Noisy Maps ...................................... 56
1.19 Pathways to Self-Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.19.1 Self-Organization Through Change of Control
Parameters ........................................ 57
1.19.2 Self-Organization Through Change of Number of
Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.19.3 Self-Organization Through Transients .. . . . . . . . . . . . . . . . 58
1.20 How We Shall Proceed .................................... 58
2. Linear Ordinary Differential Equations ..... . . . . . . . . . . . . . . . . . . . . . . 61
2.1 Examples of Linear Differential Equations: The Case of a
Single Variable ........................................... 61
2.1.1 Linear Differential Equation with Constant
Coefficient ........................................ 61
2.1.2 Linear Differential Equation with Periodic
Coefficient ........................................ 62
2.1.3 Linear Differential Equation with Quasiperiodic
Coefficient ........................................ 63
2.1.4 Linear Differential Equation with Real Bounded
Coefficient ........................................ 67
2.2 Groups and Invariance .................................... 69
2.3 Driven Systems ................................. " . . . . . . . . 72
2.4 General Theorems on Aigebraic and Differential Equations ..... 76
2.4.1 The Form ofthe Equations .......................... 76
2.4.2 Jordan's Normal Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4.3 Some General Theorems on Linear Differential
Equations .................................. . . . . . . . 77
2.4.4 Generalized Characteristic Exponents and
Lyapunov Exponents ............................... 79
2.5 Forward and Backward Equations: Dual Solution Spaces ....... 81
2.6 Linear Differential Equations with Constant Coefficients ....... 84
2.7 Linear Differential Equations with Periodic Coefficients . . . . . . . . 89
2.8 Group Theoretical Interpretation ........................... 93
2.9 Perturbation Approach* .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
