ongress information from the original version of
this book.
Library of Congress Cataloging-in-Publication Data
Scheinerman, Edward R.
Invitation to dynamical systems / Edward R. Scheinerman
p.
cm.
Includes bibliographical references and index.
ISBN 0-13-185000-8
1. Differentiable dynamical systems.
I. Title.
QA614.8.S34
1996
003.85--dc20
95-11071
CIP
All rights reserved. No part of this book may be reproduced, in any form or by any means, without
permission in writing from the author.
The names Excel, Macintosh, Maple, Mathcad, Mathematica, MATLAB, Monopoly, Mosaic, MS-DOS,
Unix, Windows, and X-Windows are trademarks or registered trademarks of their respective manufac-
turers. To Amy iv Foreword
This is the internet version of Invitation to Dynamical Systems. Unfortunately,
the original publisher has let this book go out of print. The version you are now
reading is pretty close to the original version (some formatting has changed, so page
numbers are unlikely to be the same, and the fonts are dierent).
If you would like to use this book for your own personal use, you may do so. If
you would like to photocopy this book for use in teaching a course, I will give you
my permission (but please ask). Please contact me at ers@jhu.edu. Thanks.
Please note: Some of the supporting information in this version of the book
is obsolete. For example, the description of some Matlab commands might be
incorrect because this book was written when Matlab was at version 4. In partic-
ular, the syntax for the Matlab commands ode23 and ode45 have changed in the
new release of Matlab. Please consult the Matlab documentation. The various
supporting materials (web site, answer key, etc.) are not being maintained at this
time.
Ed Scheinerman
June, 2000
v vi
Foreword Contents
Forward
v
Preface
ix
1
Introduction
1
1.1
What is a dynamical system? . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
State vectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
The next instant: discrete time . . . . . . . . . . . . . . . . .
1
1.1.3
The next instant: continuous time . . . . . . . . . . . . . . .
3
1.1.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
Mass and spring . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.2
RLC circuits . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.3
Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.4
Your bank account . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2.5
Economic growth . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2.6
Pushing buttons on your calculator . . . . . . . . . . . . . . .
14
1.2.7
Microbes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.2.8
Predator and prey . . . . . . . . . . . . . . . . . . . . . . . .
17
1.2.9
Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.2.10 Eulers method . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2.11 Random number generation . . . . . . . . . . . . . . . . . .
23
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.3
What we want; what we can get
. . . . . . . . . . . . . . . . . . . .
25
2
Linear Systems
27
2.1
One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.1
Discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.2
Continuous time . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.1.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.2
Two (and more) dimensions . . . . . . . . . . . . . . . . . . . . . . .
36
2.2.1
Discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2.2
Continuous time . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.2.3
The nondiagonalizable case*
. . . . . . . . . . . . . . . . . .
60
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.3
Examplication: Markov chains . . . . . . . . . . . . . . . . . . . . .
66
2.3.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . .
66
2.3.2
Markov chains as linear systems
. . . . . . . . . . . . . . . .
67
2.3.3
The long term
. . . . . . . . . . . . . . . . . . . . . . . . . .
69
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
vii viii
CONTENTS
3
Nonlinear Systems 1: Fixed Points
73
3.1
Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.1.1
What is a xed point? . . . . . . . . . . . . . . . . . . . . . .
73
3.1.2
Finding xed points . . . . . . . . . . . . . . . . . . . . . . .
74
3.1.3
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.2
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2.1
One dimension . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2.2
Two and more dimensions . . . . . . . . . . . . . . . . . . . .
85
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.3
Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.3.1
Linearization can fail . . . . . . . . . . . . . . . . . . . . . . .
93
3.3.2
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.3.3
Lyapunovs method
. . . . . . . . . . . . . . . . . . . . . . .
96
3.3.4
Gradient systems . . . . . . . . . . . . . . . . . . . . . . . . . 100
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4
Examplication: Iterative methods for solving equations . . . . . . . 106
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4
Nonlinear Systems 2: Periodicity and Chaos
111
4.1
Continuous time
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1.1
One dimension: no periodicity
. . . . . . . . . . . . . . . . . 111
4.1.2
Two dimensions: the Poincar´
e-Bendixson theorem . . . . . . 112
4.1.3
The Hopf bifurcation* . . . . . . . . . . . . . . . . . . . . . . 116
4.1.4
Higher dimensions: the Lorenz system and chaos . . . . . . . 118
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2
Discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2.1
Periodicity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.2
Stability of periodic points
. . . . . . . . . . . . . . . . . . . 126
4.2.3
Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2.4
Sarkovskiis theorem* . . . . . . . . . . . . . . . . . . . . . . 137
4.2.5
Chaos and symbolic dynamics . . . . . . . . . . . . . . . . . . 147
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.3
Examplication: Rie shues and the shift map . . . . . . . . . . . 159
4.3.1
Rie shues . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.3.2
The shift map . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.3.3
Shifting and shuing
. . . . . . . . . . . . . . . . . . . . . . 162
4.3.4
Shuing again and again . . . . . . . . . . . . . . . . . . . . 165
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5
Fractals
169
5.1
Cantors set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.1.1
Symbolic representation of Cantors set
. . . . . . . . . . . . 170
5.1.2
Cantors set in conventional notation . . . . . . . . . . . . . . 170
5.1.3
The link between the two representations . . . . . . . . . . . 172
5.1.4
Topological properties of the Cantor set . . . . . . . . . . . . 173
5.1.5
In what sense a fractal? . . . . . . . . . . . . . . . . . . . . . 175
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.2
Biting out the middle in the plane . . . . . . . . . . . . . . . . . . . 177
5.2.1
Sierpi´
nskis triangle
. . . . . . . . . . . . . . . . . . . . . . . 177
5.2.2
Kochs snowake . . . . . . . . . . . . . . . . . . . . . . . . . 177
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.3
Contraction mapping theorems . . . . . . . . . . . . . . . . . . . . . 180
5.3.1
Contraction maps
. . . . . . . . . . . . . . . . . . . . . . . . 180
5.3.2
Contraction mapping theorem on the real line . . . . . . . . . 181
5.3.3
Contraction mapping in higher dimensions . . . . . . . . . . . 182
5.3.4
Contractive ane maps: the spectral norm* . . . . . . . . . . 182 CONTENTS
ix
5.3.5
Other metric spaces . . . . . . . . . . . . . . . . . . . . . . . 185
5.3.6
Compact sets and Hausdor distance . . . . . . . . . . . . . . 186
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.4
Iterated function systems
. . . . . . . . . . . . . . . . . . . . . . . . 189
5.4.1
From point maps to set maps . . . . . . . . . . . . . . . . . . 190
5.4.2
The union of set maps . . . . . . . . . . . . . . . . . . . . . . 191
5.4.3
Examples revisited . . . . . . . . . . . . . . . . . . . . . . . . 193
5.4.4
IFSs dened . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.4.5
Working backward . . . . . . . . . . . . . . . . . . . . . . . . 197
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.5
Algorithms for drawing fractals . . . . . . . . . . . . . . . . . . . . . 202
5.5.1
A deterministic algorithm . . . . . . . . . . . . . . . . . . . . 202
5.5.2
Dancing on fractals . . . . . . . . . . . . . . . . . . . . . . . . 203
5.5.3
A randomized algorithm . . . . . . . . . . .