Chaos: A Program Collection for the PC

326 Pages·1999·13.617 MB·English

by  Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)

Korsch .Jodl Chaos
Springer-Verlag Berlin Heidelberg GmbH
H. J. Korsch H.-J. JodI
CHAOS
A Program Collectionfor thePC
Second Edition
With 250 Figures,
Many Numerical Experiments,
and CD-ROM for Windows 95 and NT
Springer
Professor Dr. H. J. Korsch
Professor Dr. H.-J. Jodl
Fachbereich Physik, Universität Kaiserslautern
Erwin-Schrödinger-Strasse
0-67663 Kaiserslautern, Germany
e-mail: [email protected]
[email protected]
The cover picture shows a Lorenz attractor generated with the program ODE, ordinary
differential equations, included on the CD-ROM
Additional material to this book can be downloaded from http://extras.springer.com
ISBN 978-3-662-03868-0 ISBN 978-3-662-03866-6 (eBook)
DOI 10.1007/978-3-662-03866-6
Library of Congress Cataloging-in-Publication Data applied for.
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
CHAOS: a program collection for the PC; with many numerical experimentsl
H. J. Korsch; H.-J. Jod!. -Berlin; Heidelberg; New York; London; Paris; Tokyo;
Hong Kong; Barcelona: Springer
Literaturangaben
Buch. -2. ed. -1998
CD-ROM. -2. ed. -1998
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, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication
of this publication or parts thereof is permitted only under the provisions of the German Copyright
Law of September 9, 1965, in its current version, and permission for use must always be obtained
from Springer-Verlag. Violations are hable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1994, 1999
Originally published by Springer-Verlag Berlin Heidelberg New York in 1999.
Softcover reprint of the hardcover 2nd edition 1999
Tbe 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 therefore
free for general use.
Please note: Before using the programs in this book, please consult the technical manuals provided
by the manufacturer of the computer - and of any additional plug-in boards - to be used. Tbe authors
and the publisher accept no legal responsibility for any damage caused by improper use of the instruc
tions and programs contained herein. Although these programs have been tested with extreme care, we
can offer no formal guarantee that they will function correctly. Tbe programs on the enclosed CD-ROM
are under copyright protection and may not be reproduced without written permission by Springer
Verlag. One copy of the programs may be made as a back-up, but all further copies violate copyright
law.
Cover Design: E. Kirchner, Heidelberg
Typesetting: Camera-ready copies from the authors using a Springer LATEX macro package
SPIN: 10661573 56/3144 -5 4 3 2 I 0 - Printed on acid-free paper
Preface to the Second Edition
Thestillgrowinginterest inchaoticdynamicsin physicsand thefriendly receipt
given to thefirst edition encouraged usto preparea second editionofthisbook.
During the last years, we observed an increasing introduction to chaotic
(or nonlinear) dynamics already in basiccourses in physics. Here the computer
is often used as an ideal tool for the demonstration of chaotic phenomena in
computer "experiments" during lectures. More and more students realize that
theycan benefit fromthesimultaneousinteractionwithcomputer programsand
reading oftexts,provided that specialized and easy touseprograms with many
suggestions for such "experiments" exist. The resonance from many students
and colleagues gave us the impression that our collection ofprograms helps the
students to explore the highly non-trivial behavior ofdynamical systems.
We have taken the opportunity to correct some minor errors and to clarify
a few points in the text of the book, but most of the program codes of the
first edition remained unchanged.However,the rapid development ofcomputer
operating systems made it necessary to modify some of the computer codes
and to change the installation routine. The programs in this collection were
originally written to run within the operating system DOS. Now they have
been tested to run under Windows 95and NT as well. Because of the increase
ofspeed,thecomputingtimes are noticeably reduced and much more elaborate
numerical experiments may be performed in acceptable times.
In addition to thestudents and research associates whocontributed consid
erably to the first edition, we would like to thank Dr. Martin Menzel, Dr. Leo
Schoendorffand Bernd Schellhaaf3 for their assistance in preparing this second
edition.
Kaiserslautern, H. J. Korsch and H.-J. JodI
October 1998
Preface to the First Edition
The problem, expressed in its general form, is an old one and appears under
many guises.Why are the clouds the waythey are? Is the solar system stable?
What determines the structure ofturbulence in liquids, the noise in electronic
circuits, the stability ofa plasma? What is new, that is to say with respect to
Newton's Principia published three hundred years ago,and which has emerged
over the last fewdecades, is the heuristic use ofcomputers to enhance our un
derstandingofthe mathematicsofnonlinear dynamicalprocessesand to explore
the complex behavior that even simple systems often exhibit.
It is the purpose of this book to teach chaos through a simultaneous
reading ofthe text and interaction with our selected computer programs.
The use ofcomputers is not only essential for studying nonlinear phenom
ena, but alsoenables theintuitivegeometric and heuristicapproach to be devel
oped, taught to students, and integrated into the scientist's skills, techniques,
and methods.
• Computers allow us to penetrate unexplored regions of mathematics and
discover foreseen links between ideas.
• Numerical solutions ofcomplexnonlinear problems - displayed by graphs
or videos- asopposedto analyticalsolutions,whichare often limiteddue
to the approximations made, become possible.
• The visualization of mathematics will also be a focus of this book: one
good graphorsimulationvideothat highlights the evolution ofa coherent
complex pattern can be worth more than a hundred equations.
• Nonlinear problems are almost always difficult, often having unexpected
solutions.
• In attempting to understand the details of the computer solution one
may uncover a new kind ofproblem, or a newaspect that leads to deeper
understanding.
• Appropriate graphical displays,especially those that are constructed and
composed on the screen as one interacts with the computer, will improve
the ability to choose from among promising paths. This procedure natu
rally complements the usual approaches ofexperiment, theoretical formu
lation and asymptotic approximation.
VIII Preface to the First Edition
• The benefits of the computational approach clearly depend on the avail
abilityofvariousgraphicaldisplays:smalleffectsthat maysignalnewphe
nomena (zoom); proper mappingofdatain placeofa searchfor structures
involuminousprintoutsofcolumnsofnumbers;pictureswhichclearly pro
duce an insight into the physics;color or real-time videos which enhance
perception, enabling one to correlate old and new results and recognize
unexpected phenomena.
• Interactive software in general, or educational software in physics, must
provide the ability to display in one, two or three dimensions, to show
spatial and temporal correlations, to rotate, displace or stretch objects,
to acquire diagnostic variables and essential summaries or to optimize
comparisons.
Are we providing this kind of training in our universities, so that our stu
dents may learn this method ofworkingat the 'nonlinear' frontier? We have to
find new methods to teach students to experiment with computers in the way
that we now teach them to experiment with lasers, or deepen their knowledge
of theory. Therefore, the philosophy ofour approach in this book is to practice
the use of a computer in computational physics directed at a convincing topic,
i.e. nonlinear physics and chaos. Our programs are written in such a style that
physical problems can quickly be tackled, and time is not wasted to program
detailssuch as the use ofalgorithmstosolvean equation, or the input and out
put ofdata. Of course, the student willeventually need to master the elements
of programming himselfas he comes closer to independent research.Therefore,
this book is aimed at those who have completed a course of study in physics
and are on the threshold ofresearch. Another advantage isthat 'mini-research'
can be carried out, thanks to the nature of the topic, chaos, and to the tool,
the computer. These allow one to discuss physical problems which are only
mentioned in textbooks nowadays as a potential topicofstudy,e.g. the double
pendulum.
From the point of view of the complexity of the mathematics and physics,
this book is designed mainly for students in the third or fourth year in a
science or engineering faculty.In a limited way,it might alsobe useful to those
working at the frontiers of nonlinear physics, since this topic is relatively new
and far from having well-established solutions or wide applications.
This book is organized in the following way: in Chap. 1 'Overview and
Basic Concepts', we attempt to introduce typical features of chaotic behavior
and to point out the broad applicability ofchaos in science as well as to make
the reader familiar with the terminology and theoretical concepts. In Chap.2
'Nonlinear Dynamics and Deterministic Chaos', wewilldevelop the necessary
basis, which will be deepened and applied in subsequent chapters. Chapter 3
'Billiard Systems' and Chap.4 'Gravitational Billiards' will treat two of the
'classical examples' ofsimple conservative mechanical systems. In Chap.5, the
class of different pendula, such as kicked, inverted, coupled, oscillatory or ro
tating, is representatively discussed through thedoublependulum. Phenomena
Prefaceto the First Edition IX
appearing in chaotic scattering systems are represented by the three disk scat
tering in Chap. 6. The subsequent chapters treat systems explicitly dependent
on time: namely, in Chap.7a periodically kicked particle in a box 'Fermi Ac
celeration', and in Chap.8 a driven anharmonic oscillator 'Duffing Oscillator'.
The celebrated one-dimensional iterated maps are the topic of Chap.9, and
the observed period-doubling scenario can be studied via the physical example
of nonlinear electronic circuits in Chap.10. Numerical experiments with two
dimensional maps are consideredin Chap. 11 'Mandelbrot and Julia Sets', while
Chap.12 'Ordinary Differential equations' provides a quite general platform
from which to investigate systems governed by coupled first order differential
equations. Finally, further technical questions of hardware requirements, pro
gram installation,and the use ofthe programs are addressedin the appendices.
Most chapters followthe same substructure:
- Theoretical Background
- Numerical Techniques
- Interaction with the Program
- Computer Experiments
- Real Experiments and Empirical Evidence
- References
Many books and articles have been written on chaotic experiments, and
some of the 'classical' experiments are mentioned in Chap.1. Therefore, the
last subsection in each chapter is intended to give the reader confidence to
progress from his studies on the computer to real experiments and empirical
evidence; e.g. comparing the trajectory of a double pendulum in reality and
on the screen. Of course, some aspects of the system are better studied in
the computer experiment, others in the real one; in addition,they complement
each other, e.g. looking for bifurcations in a nonlinear electronic circuit on the
oscilloscope and on the screen (Chap.10). One is at first impressed by the
apparently chaotic motion of a real double pendulum, but deeper insight into
the structure ofthis chaotic behavior isgained by looking at Poincare maps in
phase space. The experiments chosen here and briefly reported (for details see
the cited literature) are mainly for educational purposes, to be reconstructed
and used in student laboratories or in lectures. Therefore, they are not meant
to represent those experiments investigated in current nonlinear research.
The most effectivewayofusing this book may be to read a chapterwhile
working simultaneously on the computer using the appropriate program.As al
ready mentioned, the reader should use the programs - rather than program
major parts himself - in much the same way as he would use standard ser
vice software in combinationwith commercial research apparatus. On the other
hand, the use of our programs should not be a simple push button procedure,
butinvolveserious interactionwith the software.Forexample, someparameters,
X Preface to the First Edition
initial values, boundary conditions are already pre-set to execute numerical ex
perimentsdiscussed inthebook, whileothernumericalexperimentsdescribed in
detail require changes in the pre-set parameters. Further studies are suggested
and the reader may proceed independently,guided by somehints and the cited
literature. The programs are flexibleand organized in such a way that he can
set up his owncomputerexperiments, e.g.definehisownboundary in a billiard
problem or explore his favorite system ofnonlinear differential equations.
Reading this book and working with the programs requires a knowledge of
classical mechanics and a basic understandingofchaotic phenomena.The short
overview on chaotic dynamics and chaos theory in Chaps.1and 2cannot serve
as a substitute for a textbook. Within the last decade a number ofsuch books
have been published reflecting the rapid growth of the field. Some focus on
experiments, some concentrate on theory, some deal with basic concepts, while
others are simply aselection oforiginal articles. The reader should consult some
of these texts while exploring the nonlinear world by means of the computer
programs in this book.
We hope that the selected examples ofchaotic systems willhelp the reader
to gain a basic understanding ofnonlinear dynamics,and willalso demonstrate
the usefulness ofcomputers for teaching physics on the PC.
Most ofthe programs, at least in their preliminary version,were developed
by students during a seminar 'Computer Assisted Physics' (1985-1990). With
the aid oftwo grants (PPP 1987-1989, PPPP 1991-1994), wewere able to im
prove, test, standardize and update those programs. Chapter 1 contains a list
of all coauthors for every program. Wewish to acknowledge funding from the
Bundesministerium fiir Bildung und Wissenschaft (BMBW), from the Univer
sity of Kaiserslautern via the Kultusministerium of Rheinland Pfalz and the
Deutsche Forschungsgemeinschaft (DFG) for the hardware.Weare indebted to
the Volkswagenstiftung (VW) for supporting one of us (H.-J. J.) to finish this
book during a sabbatical.
Finally,wewish to recognizethe contribution ofundergraduate and gradu
ate students and research associates whoworked soenthusiastically with us on
problems associated with chaos and on the use ofcomputers in physics teach
ing.Wewouldparticularlyliketo thank the graduatestudents Bjorn Baser and
Andreas Schuch for their considerable assistance in developing and debugging
the computer codes, the instructions for interacting with the programs as well
as the large number of computer experiments. Finally, wewould like to thank
Frank Bensch and Bruno Mirbach,whoread parts ofthe manuscript and made
many useful suggestions.
Kaiserslautern, H. J. Korsch and H.-J. JodI
May 1994
Table of Contents
1 Overview and Basic Concepts 1
1.1 Introduction . 1
1.2 The Programs . 5
1.3 Literature on Chaotic Dynamics . 8
2 Nonlinear Dynamics and Deterministic Chaos 11
2.1 Deterministic Chaos . 11
2.2 Hamiltonian Systems . 12
2.2.1 Integrable and Ergodic Systems 13
2.2.2 Poincare Sections . . 16
2.2.3 The KAM Theorem ... 18
2.2.4 Homoclinic Points . . . . 20
2.3 Dissipative Dynamical Systems 21
2.3.1 Attractors . .. . 23
2.3.2 Routes to Chaos . ... 25
2.4 Special Topics . . . . . . . . . . 27
2.4.1 The Poincare-BirkhoffTheorem 27
2.4.2 Continued Fractions . 29
2.4.3 The Lyapunov Exponent . . . . 31
2.4.4 Fixed Points of One-Dimensional Maps 34
2.4.5 Fixed Points ofTwo-Dimensional Maps . 37
2.4.6 Bifurcations 42
References . . 43
3 Billiard Systems 45
3.1 Deformations ofa Circle Billiard . 47
3.2 Numerical Techniques . . . . 51
3.3 Interacting with the Program . . 52
3.4 Computer Experiments . . . . . . 56
3.4.1 From Regularity to Chaos 56
3.4.2 Zooming In . 58
3.4.3 Sensitivity and Determinism 60
3.4.4 Suggestions for Additional Experiments . 61
Stability ofTwo-Bounce Orbits .. .. . 61