Dynamics: Numerical Explorations: Accompanying Computer Program Dynamics

501 Pages·1994·15.53 MB·English

by  Helena E. Nusse

Applied Mathematical Sciences
Volume 101
Editors
F. John J.E. Marsden L. Sirovich
Advisors
M. Ohil J .K. Hale J. Keller
K. Kirchgiissner B.J. Matkowsky
J.T. Stuart A. Weinstein
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed.
2. Sirovich: Techniques of Asymptotic Analysis.
3. Hale: Theory of Functional Differential Equations, 2nd ed.
4. Percus: Combinatorial Methods.
5. von MisesiFriedrichs: F1uid Dynamics.
6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics.
7. Pipkin: Lectures on Viscoelasticity Theory.
8. Giacoglia: Perturbation Methods in Non-linear Systems.
9. Friedrichs: Spectral Theory of Operators in Hilbert Space.
10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations.
H. Wolovich: Unear Multivariable Systems.
12. Berkovitz: Optimal Control Theory.
13. Bluman/Cole: Similarity Methods for Differential Equations.
14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions.
15. Braun: Differential Equations and Their Applications, 3rd cd.
16. Lefschetz: Applications of Algebraic Topology.
17. Collatz/Wetterling: Optimization Problems.
18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol. I.
19. Marsden/McCracken: Hopf Bifurcation and Its Applications.
20. Driver: Ordinary and Delay Differential Equations.
21. Courant/Friedrichs: Supersonic F10w and Shock Waves.
22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method.
23. Lampeni: Stochastic Processes: A Survey of the Mathematical Theory.
24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II.
25. Davies: Integral Transforms and Their Applications, 2nd ed.
26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems.
27. de Boor: A Practical Guide to Splines.
28. Keilson: Markov Chain Models-Rarity and Exponentiality.
29. de Veubeke: A Course in Elasticity.
30. Shiatycki: Geometric Quantization and Quantum Mechanics.
31. Reid: Sturmian Theory for Ordinary Differential Equations.
32. Meis/Markowitz: Numerical Solution of Partial Differential Equations.
33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.
34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics.
35. Carr: Applications of Centre Manifold Theory.
36. Bengtsson/Ghil/Kiillin: Dynamic Meteorology: Data Assimilation Methods.
37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces.
38. Lichtenberg/Lieberman: Regular and Chaotic Dynamics, 2nd ed.
39. Piccini/Stampacchia/Vidossich: Ordinary Differential Equations in Rn.
40. Naylor/Sell: Unear Operator Theory in Engineering and Science.
41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors.
42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.
43. Ockendon/Taylor: Inviscid F1uid F1ows.
44. Pazy: Semigroups of Un~r Operators and Applications to Partial Differential Equations.
45. GlashoffiGustafson: Unear Operations and Approximation: An Introduction to the Theoretical Analysis
and Numerical Treatment of Semi-Infinite Programs.
46. Wilcax: Scattering Theory for Diffraction Gratings.
47. Hale et al: An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory.
48. Murray: Asymptotic Analysis.
49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics.
50. Wi/cax: Sound Propagation in Stratified Fluids.
51. Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol. I.
(continued following index)
Helena E. Nusse James A. Yorke
Dynamics:
Numerical Explorations
Accompanying Computer Program Dynamics
Coauthored by Eric J. Kostelich
With 198 Illustrations, 8 in Color, and a 3 Y2 " DOS Diskette
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Helena E. Nusse James A. Yorke
Vakgroep Econometrie Institute for Physical Science and
Rijksuniversiteit Groningen Technology
NL-9700 A V Groningen University of Maryland
The Netherlands College Park, MD 20742
and USA
Institute for Physical Science and
Technology Coauthor of Dynamics
University of Maryland Eric J. Kostelich
College Park, MD 20742 Department of Mathematics
USA Arizona State University
Tempe, AZ 85287
USA
Editors
F. John J .E. Marsden L. Sirovich
Courant Institute of Department of Mathematics Division of Applied
Mathematical Sciences University of California Mathematics
New York University Berkeley, CA 94720 Brown University
New York, NY 10012 USA Providence, RI 02912
USA USA
Mathematics Subject Classification (1991): 49L20, 58028, 70K15, 90C39
On the cover: Basin, stable and unstable manifold, and a straddle trajectory. For more details, see Figure
9-5b in the text.
Library of Congress Cataloging-in-Publication Data
Nusse, Helena Engelina, 1952-
Dynamics: numerical explorations/Helena E. Nusse, James A. Yorke.
p. em. - (Applied mathematical sciences; v. 101)
Includes bibliographical references (p. - ) and index.
ISBN 0-387-94254-8 (New York). - ISBN 3-540-94254-8 (Berlin): DM98.00
1. Dynamics. 2. Dynamics -Data processing. 3. Chaotic behavior
in systems-Data processing. I. Yorke, James A. II. Title.
Ill. Series: Applied mathematical sciences (Springer-Verlag New York Inc.); v. 101.
QA1.A647 vol. 101
[QA845]
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[003' .851 93·50710
Printed on acid-free paper.
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DOl: 10.1007/978-1-4684-0231-5
Preface
Plotting trajectories is a useful capability in exploring a dynamical
system, but it is just the beginning. The Maryland Chaos Group developed an
array of tools to help visualize the properties of dynamical systems
induding automatic method for plotting all "basins and attractors ",
computing "straddle trajectories", and for automatically searching for all
periodic orbits of a specified period.
In the investigations of the Maryland Chaos Group, I.A. Yorke found it
useful to be able to combine these various basic tools with each other into
a single package that grew with time so that each new study could benefit
from the previous programming efforts. He has been writing this software
and distributing versions for the last nine years. The resulting program
Dynamics requires either a Unix workstation running XII graphics or an IBM
PC compatible computer. Eric I. Kostelich has put in a great deal of effort
to port the program to Unix workstations. Some basic tools in Dynamics,
such as the computation of Lyapunov exponents and the use of Newton's
method are standard. The method of computation of stable and unstable
manifolds is superior to standard procedures. Dynamics is currently being
used extensively in our research and it is being used in undergraduate
courses.
Dynamics: Numerical Explora#ons provides an introduction to and
overview of fundamental tools and numerical methods together with many
simple examples. All the numerical methods described in this book are
implemented in Dynamics. While the algorithms to implement these ideas are
sometimes fairly sophisticated, they are elementary in that what they do
can be understood by undergraduates. They should be available to everyone
exploring dynamical systems. Many of the examples reveal patterns that are
not fully understood and have surprises lurking just beyond the edges of
the imagination. This package can be used by undergraduates, by graduate
students, and by researchers in a variety of scientific disciplines.
Improving the program is a continuous process so there may be
improvements in the program not reflected in the handbook. In such cases,
these changes are self explanatory.
Preface v
An overview of capabilities of Dynamics
This program will continue to evolve. It is a toolkit in which the
tools are· all available at any moment, enabling you to explore the system
with much greater ease than if each tool was a separate program. These
tools are elementary and should be available to everyone exploring
dynamical systems. Some of the tools have not previously been available
anywhere, indicating only that the ideas used in implementing them are not
obvious. Nonetheless, the capabilities presented here are quite basic.
• The program iterates maps and solves differential equations. The
program utilizes fixed and variable step size Runge-Kutta solvers for
differential equations.
• Trajectories can be plotted and you can interactively store results
and change initial conditions, parameters, and the scale of the screen.
• If desired, the user can split the screen into quadrants. The
different quadrant windows can have different coordinates being plotted,
with simultaneous plotting in all windows. Figure I-Ion page 3 is an
example of a screen divided into quadrants.
• The program features an array of simple commands. They can be
invoked while a complicated process is being carried out. Some examples
are: the screen can be cleared or refreshed; crosses can be plotted; the
system can be paused or "single stepped" one point at a time; current
positions can be stored; a trajectory can be reinitialized, to mention a
few. The arrow keys may be used for drawing boxes, rescaling the screen, or
choosing different initial points. The images of the small cross can be
plotted, and as the user moves the small cross using the arrow keys the
program continues to show the images of the small cross.
• The state of the program can be stored for later use. The program
can create a file of parameters that have been set, and this file of values
and settings can automatically be reinstalled later when restarting the
program.
• Pictures created on the screen can be printed in resolution higher
than that of the screen (960 dots wide by 544 vertically or the resolution
720 by 720 which is used throughout this handbook), and these images can be
stored on a disk (in a data compressed format). Pictures can be recalled
from a disk and added to each other. Currently supported printers are the
Epson printer MX-80 printer, the Epson QL 2500 color printer, Hewlett
Packard LaserJet printers, and printers compatible with these. It also
supports PostScript printers. The PostScript printer support was added by
Eric J. Kostelich. (All the pictures in this book were made by HP LaserJet
printers.)
vi Dynamics: Numerical Explorations
An overview of advanced capabilities of Dynamics
With this program you can:
• Find fixed points and periodic orbits using Quasi-Newton's method
and (when the system is two dimensional) find the eigenvalues and
eigenvectors of the derivatives of the processes evaluated at the periodic
orbits.
• Calculate Lyapunov exponents and the Lyapunov dimension of an
attractor.
• Automatically plot all basins of attractors and attractors for 2-
dimensional processes.
• Follow periodic orbits as a parameter is varied. Attracting and
unstable orbits are plotted in different colors.
• Compute straddle orbits that are chaotic but do not lie on a chaotic
attractor. Compute a bounded chaotic trajectory of the Henon map when
almost all initial points diverge to infinity.
• Plot unstable and stable manifolds of periodic orbits.
• Create bifurcation diagrams showing how attractors change as a
parameter is varied.
• Automatically find and plot the periodic orbits of a specified
period.
Help for the novice
Help files are available on-line. There is a menu of help facilities
and the program provides an on-line quick start tutorial for the beginner.
Examples and exercises
Chapter 2 presents examples of pictures you can make simply. The
required commands are printed in bold. In Chapters 5 through II, we
present more examples for creating reliable pictures. You are invited to
make the exercises to get familiar with the majority of the features of
Dynamics. A few exercises are preceded with a '*', and are considered to be
rather difficult.
Topics of discussion
Below many of the figures, we suggest a "topic of discussion" and pose
a question concerning some feature of the figure. Sometimes there is a
simple answer but more often there are a variety of correct possible
answers, and sometimes there are obvious extensions. For such questions,
the exchange of ideas in discussions is beneficial to all.
Preface vii
References
In Chapters 5 through 11 we include a section entitled "References
related to Dynamics". The purpose of this section is twofold. One purpose
is to include references that establish the reliability of the numerical
methods of the program. A second one is to illustrate how Dynamics is used
or can be used.
At the end of this handbook we give a selection of references. In this
list, we have obviously left out many important contributions to the field
of dynamical systems. Consult, for example, the 4405 references in Shiraiwa
(1985) and the 7157 references in Zhang (1991).
Disclaimer
While most of the routines have been tested during a period of years,
they are not designed for commercial application. The authors and the
publisher assume no responsibility for losses that might result from errors
in this program. Comments, questions, and suggestions on the software
package can be directed to H.E.N. (book) and J.A.Y. (program). However, we
cannot give help or replies in all cases.
Acknowledgments
We would like to thank the many people who have made useful comments
on the program and this handbook. We would like to thank all the
researchers of the Maryland Chaos Group for their comments. In particular,
we would like to thank Kevin Duffy, Jason Gallas, Olaf Harnmelburg, Brian
Hunt, Hiiseyin Koc;:ak, Ajay Kochhar, Tim Sauer, and Paul Schure for their
comments and suggestions.
The basic research reflected in this program has been supported in
part by the Air Force Office of Scientific Research/Applied Mathematics,
the Department of Energy (Scientific Computing Staff Office of Energy
Research), the Office of Naval Research/Defense Advanced Research Projects
Agency/Applied and Computational Mathematics Program, and the National
Science Foundation Computational Mathematics Program.
The file Dynamics. exe was compressed using LZEXE (Version 0.91,
Copyright (c) Fabrice Bellard 1989.) The archives Cfiles.exe and
OBJflles.exe were created using LHA (Version 2.12, Copyright (c) Haruyasu
Yoshizaki 1988-1991). IBM is a registered trademark of International
Business Machines, QuickC and MS-DOS are registered trademarks of
Microsoft, Unix is a registered trademark of American Telegraph & Telephone
company, and PostScript printer is a registered trademark of Adobe Systems
Inc. ChiWriter 4.1 was used for typesetting the text of this book.
Helena E. Nusse and James A. Yorke
College Park
viii Dynamics: Numerical Explorations
Contents
Preface v
Color Pictures xiii
*Color Insert is between pp. 314 and 315.
1
1. Getting the program running
1.1 The Dynamics program and hardware 1
1.2 Getting started with Dynamics 7
16
Appendix: Description of the interrupts
19
2. Samples of Dynamics: pictures you can make simply
19
2.1 Introduction
35
2.2 Complex pictures that are simple to make
Appendix: Command for plotting a graph and
Commands from the Main Menu 123
125
3. Screen utilities
3.1 Clear or Refresh screen and set Text Level
125
(Screen Menu SM)
3.2 The arrow keys and boxes
129
(BoX Menu BXM)
3.3 Initializing trajectories, plotting crosses,
drawing circles and their iterates
135
(cross Menu KM)
3.4 Drawing axes
(AXes Menu AXM) 142
3.5 Windows and rescaling
(Window Menu WM) 147
3.6 Setting colors
(Color Menu CM and Color Table Menu CTM) 153
Contents ix
4. Utilities 167
4.1 Setting parameters
(Parameter Menu PM) 167
4.2 Setting and replacing a vector
(Vector Menu VM) 176
4.3 Setting step size
(Differential Equation Menu DEM) 181
4.4 Saving pictures and data
(Disk Menu DM) 188
4.5 Setting the size of the core
(Size of Core Menu SCM) 194
4.6 Printing pictures
(PriNter Menu PNM) 196
5. Dimension and Lyapunov exponents 201
5.1 Introduction and the Methods 201
5.2 Lyapunov Menu LM 216
5.3 Examples 222
5.4 Exercises 226
5.5 References related to Dynamics 228
6. Bifurcation diagrams 229
6.1 Introduction and the Methods 229
6.2 BIFurcation diagram Menu BIFM 240
6.3 Examples 249
6.4 Exercises 262
6.5 References related to Dynamics 268
7. Basins of attraction 269
7.1 Introduction and the Methods 269
7.2 Basin of attraction Menu BM 282
7.3 Examples 295
7.4 Exercises 308
7.5 References related to Dynamics 312
8. Straddle trajectories 315
8.1 Introduction and the Methods 315
8.2 Straddle Trajectory Menu STM 328
8.3 Examples 336
8.4 Exercises 342
8.5 References related to Dynamics 346
x Dynamics: Numerical Explorations