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.
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. Giittinger and H. Eikemeier
Volume 5 Pattern Formation 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, 1. 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 Chemical 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 ofInteracting 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 Chemical Oscillations, Waves, and Turbnlence By Y Kuramoto
Volume 20 Advanced Synergetics By H. Haken
Volume 21 Stochastic Phenomena and Chaotic Behaviour in Complex Systems
Editor: P. Schuster
Volume 22 Synergetics - From Microscopic to Macroscopic Order Editor: E. Frehland
Volume 23 Synergetics of the Brain Editors: E. Ba~ar, H. Flohr, H. Haken, and AJ. Mandell
Volume 24 Chaos and Statistical Methods Editor: Y. Kuramoto
Stochastic Phenomena
and Chaotic Behaviour
in Complex Systems
Proceedings of the Fourth Meeting of the
UNESCO Working Group on Systems Analysis
Flattnitz, Kiirnten, Austria, June 6-10, 1983
Editor: P. Schuster
With 108 Figures
Springer.:verlag
Berlin Heidelberg New York Tokyo 1984
Professor Dr. Peter Schuster
Institut fdr Theoretische Chemie und Strablenchemie der Universitiit Wien,
Wiihringer StraBe 17
A-I090 Wien, Austria
Series Editor:
Professor Dr. Dr. h. c. Hermann Haken
Institut fUr Theoretische Physik der Universitiit Stuttgart, PfafTenwaldring S7/IV,
D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN-13 :978-3-642-69593-3 e-ISBN-13 :978-3-642-69591-9
001: 10.1007/978-3-642-69591-9
Library of Congress Cataloging in Publication Data. Main entry under title: Stochastic phenomena and
chaotic behaviour in complex systems. (Springer series in synergetics ; v. 21). "Workshop was sponsored by
UNESCO, Paris and the Bundesministerium flir Wissenschaft und Forschung, Wien" - Pref. 1. Chaotic
behavior in systems-Congresses. 2. Stochastic processes-Congresses. I. Schuster, P. (peter), 1941-. II. Unesco.
Ill. Austria. Bundesministerium flir Wissenschaft und Forschung. Iv. Series. QA402.S8475 1984 003
84-1251
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© by Springer-Verlag Berlin Heidelberg 1984
Softcover reprint of the hardcover lst edition 1984
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2153/3130-543210
Preface
This book contains all invited contributions of an interdisciplinary workshop of
the UNESCO working group on systems analysis of the European and North American
region entitled "Stochastic Phenomena and Chaotic Behaviour in Complex Systems".
The meeting was held at Hotel Winterthalerhof in Flattnitz, Karnten, Austria from
June 6-10, 1983.
This workshop brought together some 20 mathematicians, physicists, chemists,
biologists, psychologists and economists from different European and American coun
tries who share a common interest in the dynamics of complex systems and their ana
lysis by mathematical techniques. The workshop in Flattnitz continued a series of
meetings of the UNESCO working group on systems analysis which started in 1977 in
Bucharest and was continued in Cambridge, U.K., 1981 and in Lyon, 1982.
The title of the meeting was chosen in order to focus on one of the current
problems of the analysis of dynamical systems. A deeper understanding of the vari
ous sources of stochasticity is of primary importance for the interpretation of
experimental observations. Chaotic dynamics plays a central role since it intro
duces a stochastic element into deterministic systems.
The workshop was sponsored by UNESCO, Paris and the Bundesministerium fUr
Wissenschaft und Forschung, Wien. We are greatly indebted to the Bundesminister fUr
Wissenschaft und Forschung, Dr. Herta Firnberg and to the Head of the Scientific
Cooperation Bureau for the European and North American Region, Prof. Dr. J. Jaz,
who made this workshop possible through their financial support. The warm hospita
lity of Familie Klimbacher and the staff of Hotel Winterthalerhof is gratefully
acknowledged. We thank Mrs. J. Jakubetz, Mag. B. Gassner, Mag. M. Reithmaier, Dr.
F. Kemler and Mr. J. Konig for their technical assistence in organizing the meeting
and preparing the proceedings. In addition, our thanks go to Professor H. Haken for
having these Proceedings included in the Springer Series in Synergetics.
Wien
November 1983 Peter Schuster
v
Contents
Introductory Remarks
By P. Schuster ...............•..............................................
Part I General Concepts
Some Basic Ideas on a Dynamic Information Theory
By H. Haken (With 3 Figures) ...............•................................ 6
Aspects of Optimization and Adaptation
By Y.M. Ermoliev ................•.•.....•................................... 13
Relaxed Markov Processes, Jackson Networks and Polymerisation
By P. Whittle ............................................................... 17
Part 1/ Chaotic Dynamics-Theory
Noodle-Map Chaos: A Simple Example
By O.E. Rossler, J.L. Hudson, and J.D. Farmer (With 1 Figure) ....•.•........ 30
A Mechanism for Spurious Solutions of Nonlinear Boundary Value Problems
By H.-O. Peitgen (With 6 Figures) ...........•...................•...•.•.•... 38
Approach to Equilibrium: Kuzmin's Theorem for Dissipative and Expanding Maps
By D. Mayer ......................•.......•.................•.•...•.•.......• 52
Complex Behaviours in Macrosystems Near Polycritical Points
By P. Coullet (With 6 Figures) ...........•.................................. 67
Part 1/1 Chaotic Dynamics - Real Systems and Experimental Verification
Chaos in Classical Mechanics: The Double Pendulum
By P.H. Richter and H.-J. Scholz (With 9 Figures) 86
Chaos in Continuous Stirred Chemical Reactors
By J.L. Hudson, J.C. Mankin, and O.E. Rossler (With 4 Figures) ...•. ,........ 98
The Interface Between Mathematical Chaos and Experimental Chemistry
By R.M. Noyes (With 4 Figures) .............................................. 106
The Enzyme and the Strange Attractor -Compari sons of Experimental and
Numerical Data for an Enzyme Reaction with Chaotic Motion
ByL.F. Olsen (\~ith 7 Figures) ...•...................•.•...........•.•...... 116
VII
Nonuniform Information Processing by Strange Attractors of Chaotic Maps
and Flows
By J.S. Nicolis, G. Mayer-Kress, and G. Haubs (With 13 Figures) ., ........... 124
Part IV Stability and Instability in Dynamical Networks
Generalized Modes and Nonlinear Dynamical Systems
By P.E. Phillipson (With 3 Figures) ......................................... 142
Dynamics of Linear and Nonlinear Autocatalysis and Competition
By J. Hofbauer and P. Schuster (With 6 Figures) ........•...............•.... 161
Permanence and Uninvadability for Deterministic Population ~lodels
By K. Sigmund and P. Schuster (With 2 Figures) ...........................•. , 173
Part V Stochasticity in Complex Systems
Random Selection and the Neutral Theory-Sources of Stochasticity in
Replication
By P. Schuster and K. Sigmund (Hith 9 Figures) .............................. 186
The Dynamics of Catalytic Hypercycl es -A Stochastic Simulation
By A.M. Rodriguez-Vargas and P. Schuster (With 6 Figures) ................... 208
Perturbation-Dependent Coexistence and Species Diversity in Ecosystems
By M. Rejmanek (Hith 7 Figures) ............................................. 220
Chance and Necessity in Urban Systems
By P.M. Allen, M. Sanglier, and G. Engelen (With 14 Figures) 231
Random Phenomena in Nonlinear Systems in Connection with the Volterra Approach
By K.F. Albrecht, V. Chetverikov, W. Ebeling, R. Funke, W. Mende, and
r·1. Peschel (vJith 8 Figures) ................................................. 250
List of Contributors 271
VIII
Introductory Remarks
P. Schuster
Institut fUr Theoretische Chemie und Strahlenchemie, Universitat Wien
Wahringer StraBe 17, A-1090 Wien, Austria
The notions "strange attractor", "chaotic dynamics" or simply
"chaos" are synonyms for a kind of complicated dynamical behaviour
with two main features: (1) The long-time dynamics is characterized
by irregular changes in variables which are lacking any periodicity
or quasiperiodicity (2) trajectories through close by lying points
diverge exponentiall~-with time. The search for strange attractors
in nonlinear dynamical systems became kind of a' fashion after
LORENZ [1] had published his famous work on a model differential
equation for hydrodynamic flow which shows chaotic dynamics for
certain choices of parameters. Indeed, most people were successful
and found chaotic solutions for various nonlinear difference and
differential equations provided the number of independent variables
was high enough and the nonlinearities were sufficiently general.
It took much longer to work out the deeper mathematical properties
of chaos, its physical meaning is not very well understood yet. For
some publications concerning current problems of the research on
strange attractors and their properties see [2-4]. In this volume
the contribution by ROESSLER deals with the problem of classificat
ion of strange attractors, MAYER presents an investigation on in
variant measures for exponentially expanding maps.
Chaos introduces an interface between determinism and randomness as
FORD [5] points out in a well-written recent article. Truly deter
ministic description of chaotic dynamics requires infinite precision
in the choice of initial conditions and thus, is a scientific
chimera. Therefore, chaos introduces a fundamental uncertainty which
is more general than HEISENBERG's uncertainty in quantum mechanics:
it concerns also macroscopic dynamics and restricts the available
information for every variable whereas the quantum mechanical un
certainty operates on canonically conjugate pairs of variables only.
The present state of affairs in the description of real systems with
chaotic dynamics calls for a new approach towards a more complete
description of experiments. Such a description takes into account
explicitly the inevitable uncertainties by means of an appropriate
mathematical formalism and unites thereby deterministic kinetics
and the various sources of stochasticity. A mathematical tool for
such an approach in available, at least in principle: almost all
real dynamical systems in continuous time can be described properly
by a master equation. The problem actually is the search for solut
ions which most often is extremely hard and successful in except
ional cases only. New methods of approximation were developed
during the last decade [6,7] but they are not applicable to the
complex dynamics we are basically interested in here.
Difference equations of very simple algebraic structures may have
extremely complicated dynamical behaviour. The discretized logistic
equation is one of the most popular examples of this kind.(For an
easy to read and recent review see [8]). The very rich dynamics
introduced by the choice of discrete time may also cause serious
problems for the numerical approach: numerical integration inevit
ably is based on discrete time and can result in highly complicated
spurious solutions as the contribution by PEITGEN nicely demonstr
ates.
How common is chaotic dynamics in systems with continuous time? In
order to find an answer we divide dynamical systems into two
classes:
(1) Systems which are derived from "true" equations of motion and
hence in general have a non-zero kinetic energy:
+ ( 1 ) •
In systems of this kind chaotic dynamics leading to ergodic behav
iour is very frequent. Separable Hamiltonian systems charac~erized
by very simple dynamics are rare in reality although they make up
for most examples treated in elementary texts of classical mechan
ics. The contribution by RICHTER and SCHOLZ dealing with chaotic
dynamics found with the double pendulum is a characteristic example.
Closely related are problems in hydrodynamics: chaotic solutions
of the NAVIER-STOKES equation are discussed in relation to turbul
ence [3,9].
(2) Systems which can be derived formally from (1) by extrapolation
to zero mass (or zero kinetic energy) 1). An alternative interpret-
1)Such an extrapolation is not trivial since one boundary condition
is lost thereby as FRIEDRICHS [10] pointed out some time ago. It
has to be performed with sufficient care.
2
ation of the neglect of the second derivatives is motion at very
high friction, i.e. motion in a medium of high viscosity (high
values of y). Equations of this truncated type
dXk
dt = Gk (x1,···,xn) ~ k=1, ••• ,n (2)
are commonly considered in chemical kinetics of homogeneous
solutions, i.e. when diffusion can be neglected or in well stirred
reactors, and in many other disciplines like mathematical ecology,
population genetics, game dynamics, dynamical economies. In such
systems with zero kinetic energy strange attractors are common
when Gk is sufficiently non-linear and k~3. In almost all well
studied examples the occurrence of chaotic dynamics is confined to
very small regions in parameter space. Detecting these regions re
quires mathematical intuition and numerical skill. New methods are
being developed for this purpose, one example is presented by
COULLET in his contribution. Needless to say, the experimental veri
fication of chaotic dynamics in systems of this kind is an extremely
hard problem: how to distinguish "true" chaos in the solution of
the differential equations and irregular fluctuations of external
parameters within the range of control ? The highly precise and
elegant experimental and numerical works by HUDSON, MANKIN,
ROESSLER and NOYES as well as their scientific dispute serve as an
illustrative demonstration of the enormous difficulties to be
encountered in this new branch of chemical kinetics. OLSEN's
contribution presents an example for the occurrence of a strange
attractor in an enzyme-catalysed reaction. NICOLIS, MAYER-KRESS
and HAUBS discuss the role of chaotic dynamics in the operation
of an information processor,particularly the human brain.
The contributions in Part One deal with new general concepts in
systems analysis: a dynamical information theory by HAKEN, an
approach to describe optimization and adaptation in fluctuating
systems by ERMOLIEV and a contribution on relaxed Markov chains
by WHITTLE.
One part of these proceedings contains three papers dealing with
the search for stability and instability in non-linear dynamical
systems. PHILLIPSON presents a new method to construct analytical
approximations to the solutions of ordinary differential equations
which is shown to be useful as diagnostics of stability.
The two contributions by HOFBAUER, SIGMUND and SCHUSTER deal with
biochemical and biological examples of stability analysis in non
linear dynamical systems.
3
