Chaos and fractals: an elementary introduction
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2012
|
| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXI, 408 S. Ill., graph. Darst. |
| ISBN: | 9780199566440 9780199566433 |
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| 245 | 1 | 0 | |a Chaos and fractals |b an elementary introduction |c David P. Feldman |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2012 | |
| 300 | |a XXI, 408 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
I Introducing Discrete Dynamical Systems
1
0
Opening Remarks
3
0.1
Chaos
3
0.2
Fractals
4
0.3
The Character of Chaos and Fractals
5
1
Functions
9
1.1
Functions as Actions
9
1.2
Functions as a Formula
10
1.3
Functions are Deterministic
11
1.4
Functions as Graphs
11
1.5
Functions as Maps
13
Exercises
14
2
Iterating Functions
17
2.1
The Idea of Iteration
17
2.2
Some Vocabulary and Notation
18
2.3
Iterated Function Notation
19
2.4
Algebraic Expressions for Iterated Functions
20
2.5
Why Iteration?
21
Exercises
23
3
Qualitative Dynamics: The Fate of the Orbit
25
3.1
Dynamical Systems
25
3.2
Dynamics of the Squaring Function
26
3.3
The Phase Line
27
3.4
Fixed Points via Algebra
27
3.5
Fixed Points Graphically
29
3.6
Types of Fixed Points
30
Exercises
31
4
Time Series Plots
33
4.1
Examples of Time Series Plots
33
Exercises
35
5
Graphical Iteration
37
5.1
An Initial Example
37
5.2
The Method of Graphical Iteration
38
5.3
Further Examples
39
Exercises
42
xvi Contents
Iterating
Linear
Functions
45
6.1
A Series of Examples
45
6.2
Slopes of
+1
or
-1 48
Exercises
50
Population Models
53
7.1
Exponential Growth
53
7.2
Modifying the Exponential Growth Model
56
7.3
The Logistic Equation
59
7.4
A Note on the Importance of Stability
62
7.5
Other
r
Values
64
Exercises
65
Newton, Laplace, and Determinism
67
8.1
Newton and Universal Mechanics
67
8.2
The Enlightenment and Optimism
69
8.3
Causality and Laplace s Demon
70
8.4
Science Today
71
8.5
A Look Ahead
72
II Chaos
75
9
Chaos and the Logistic Equation
77
9.1
Periodic Behavior
77
9.2
Aperiodic Behavior
82
9.3
Chaos Defined
85
9.4
Implications of Aperiodic Behavior
86
Exercises
87
10
The Butterfly Effect
89
10.1
Stable Periodic Behavior
89
10.2
Sensitive Dependence on Initial Conditions
90
10.3
SDIC Defined
93
10.4
Lyapunov Exponents
95
10.5
Stretching and Folding: Ingredients for Chaos
97
10.6
Chaotic Numerics: The Shadowing Lemma
99
Exercises
102
11
The Bifurcation Diagram
105
11.1
A Collection of Final-State Diagrams
105
11.2
Periodic Windows HO
11.3
Bifurcation Diagram Summary
Ш
Exercises
112
12
Universality
115
12.1
Bifurcation Diagrams for Other Functions
П5
12.2
Universality of Period Doubling
118
12.3
Physical Consequences of Universality
121
Contents xvii
12.4 Renormalization
and Universality
124
12.5
How are Higher-Dimensional Phenomena Universal?
128
Exercises
129
13
Statistical Stability of Chaos
131
13.1
Histograms of Periodic Orbits
131
13.2
Histograms of Chaotic Orbits
132
13.3
Ergodicity
135
13.4
Predictable Unpredictability
138
Exercises
139
14
Determinism, Randomness, and Nonlinearity
141
14.1
Symbolic Dynamics
141
14.2
Chaotic Systems as Sources of Randomness
143
14.3
Randomness?
144
14.4
Linearity, Nonlinearity, and Reductionism
148
14.5
Summary and a Look Ahead
152
Exercises
154
III Fractals
155
15
Introducing Fractals
157
15.1
Shapes
157
15.2
Self-Similarity
158
15.3
Typical Size?
160
15.4
Mathematical vs. Real Fractals
161
Exercises
162
16
Dimensions
163
16.1
How Many Little Things Fit inside a Big Thing?
163
16.2
The Dimension of the Snowflake
165
16.3
What does
D
« 1.46497
Mean?
166
16.4
The Dimension of the Cantor Set
167
16.5
The Dimension of the
Sierpiński
Triangle
168
16.6
Fractals, Defined Again
169
Exercises
170
17
Random Fractals
173
17.1
The Random Koch Curve
173
17.2
Irregular Fractals
176
17.3
Fractal Landscapes
178
17.4
The Chaos Game
178
17.5
The Role of Randomness
181
17.6
The Collage Theorem
181
Exercises
184
18
The Box-Counting Dimension
187
18.1
Covering a Box with Little Boxes
187
xviii Contents
18.2
Covering a Circle with Little Boxes
189
18.3
Estimating the Box-Counting Dimension
190
18.4
Summary
193
Exercises
193
19
When do Averages Exist?
195
19.1
Tossing a Coin
195
19.2
St. Petersburg Game
198
19.3
Average Winnings for the St. Petersburg Game
202
19.4
Implications
203
Exercises
204
20
Power Laws and Long Tails
207
20.1
The Central Limit Theorem and Normal Distributions
207
20.2
Power Laws: An Initial Example
211
20.3
Power Laws and the Long Tail
213
20.4
Power Laws and Fractals
215
20.5
Where do Power Laws Come From?
218
Exercises
219
21
Infinities, Big and Small
221
21.1
What is the Size of the Cantor Set?
221
21.2
Cardinality, Counting, and the Size of Sets
222
21.3
Countable Infinities
224
21.4
Rational and Irrational Numbers
225
21.5
Binary
225
21.6
The Cardinality of the Unit Interval
227
21.7
The Cardinality of the Cantor Set
229
21.8
Summary and a Look Ahead
232
Exercises
232
IV Julia Sets and the Mandelbrot Set
235
22
Introducing Julia Sets
237
22.1
The Squaring Function
237
22.2
Other Examples
238
22.3
Summary
239
Exercises
240
23
Complex Numbers
241
23.1
The Square Root of
-1 241
23.2
The Algebra of Complex Numbers
242
23.3
The Geometry of Complex Numbers
243
23.4
The Geometry of Multiplication
244
Exercises
246
24
Julia Sets for the Quadratic Family
249
24.1
The Complex Squaring Function
249
Contents xix
24.2
Another Example:
f (z)
=
z2
- 1 250
24.3
Julia
Sets
for
ƒ
(z)
=
z2
+
с
252
24.4
Computing and Coloring Julia Sets
253
Exercises
255
25
The Mandelbrot Set
257
25.1
Cataloging Julia Sets
257
25.2
The Mandelbrot Set Defined
258
25.3
The Mandelbrot Set and the Critical Orbit
259
25.4
Exploring the Mandelbrot Set
260
25.5
The Mandelbrot Set is a Julia Set Encyclopedia
263
25.6
Conclusion
268
Exercises
269
V Higher-Dimensional Systems
271
26
Two-Dimensional Discrete Dynamical Systems
273
26.1
Review of One-Dimensional Discrete Dynamics
273
26.2
Two-Dimensional Discrete Dynamical Systems
274
26.3
The
Hénon Map
275
26.4
Chaotic Behavior and the
Hénon Map
277
26.5
A Chaotic Attractor
279
26.6
Strange Attractors Defined
283
Exercises
285
27
Cellular Automata
287
27.1
One-Dimensional Cellular Automata: An Initial Example
287
27.2
Surveying One-Dimensional Cellular Automata
290
27.3
Classifying and Characterizing
С
A Behavior
293
27.4
Behavior of CAs Using a Single-Cell Seed
296
27.5
CA Naming Conventions
298
27.6
Other Types of CAs
300
Exercises
302
28
Introduction to Differential Equations
303
28.1
Continuous Change
303
28.2
Instantaneous Rates of Change
304
28.3
Approximately Solving a Differential Equation
306
28.4
Euler s Method
310
28.5
Other Solution Methods
311
Exercises
312
29
One-Dimensional Differential Equations
313
29.1
The Continuous Logistic Equation
313
xx Contents
29.2
Another Example
316
29.3
Overview of One-Dimensional Differential Equations
317
Exercises
318
30
Two-Dimensional Differential Equations
321
30.1
Introducing the
Lotka-
Volterra Model
321
30.2
Euler s Method in Two Dimensions
323
30.3
Analyzing the
Lotka-
Volterra Model
325
30.4
Phase Space and Phase Portraits
326
30.5
Another Example: An Attracting Fixed Point
329
30.6
One More Example: Limit Cycles
330
30.7
Overview of Two-Dimensional Differential Equations
331
Exercises
332
31
Chaotic Differential Equations and Strange Attractors
335
31.1
The
Lorenz
Equations
335
31.2
A Fixed Point
336
31.3
Periodic Behavior
338
31.4
Chaos and the
Lorenz
Equations
339
31.5
The
Lorenz Attractor 342
31.6
The
Rössler
Attractor
345
31.7
Chaotic Flows and One-Dimensional Functions
347
Exercises
349
VI Conclusion
351
32
Conclusion
353
32.1
Summary
353
32.2
Order and Disorder
354
32.3
Prediction and Understanding
355
32.4
A Theory of Forms
355
32.5
Revolution or Reconfiguration?
357
VII
Appendices
361
A Review of Selected Topics from Algebra
363
A.I Exponents
363
A.2 The Quadratic Formula
366
A.3 Linear Functions
368
A.
4
Logarithms
370
Exercises
374
В
Histograms and Distributions
377
B.I Representing Data with Histograms
377
B.2 Choosing Bin Sizes
378
B.3 Normalizing Histograms
381
B.4 Approximating Histograms with Functions
383
Exercises
386
С
Suggestions
for Further Reading
C.I (Mostly) Books
C.2 Peer-Reviewed Papers
C.3 Suggestions for Further Reading
References
Index
Contents
xxi
389
389
393
395
397
403
This book provides the reader with an elementary introduction to chaos and
fractals, suitable for students with a background in elementary algebra, without
assuming prior coursework in calculus or physics. Simple iterated functions are used to
introduce the key phenomena of chaos: aperiodicity, bifurcations, and sensitive dependence on
initial conditions, also known as the butterfly effect. Fractals are introduced as self-similar geometric
objects and analyzed with the self-similarity and box-counting dimensions. After a brief discussion of power
laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines cellular
automata, chaotic differential equations, and strange attractors.
The book is richly illustrated and includes over
250
end-of-chapter exercises. A flexible format and a clear
and succinct writing style make it a good choice for introductory courses in chaos and fractals.
David P. Feldman is Professor of Physics and Mathematics at College of the Atlantic, Bar Harbor, Maine, USA.
Chaos and
f
ratìats are
two intertwined concepts that have revolutionized many areas of science and renewed
popular interest in mathematics over the past few decades. Feldman s book is a rich resource for anyone who
wants a deeper understanding of these subjects without the need for advanced mathematics.
Julien
Clinton Sprott, Emeritus Professor of Physics, University of Wisconsin-Madison
David P. Feldman provides a delightful and thoughtful
introduďion
to chaos and fractals requiring only a
good background in algebra. The formal treatment of nonlinear dynamics, chaotic behavior, Lyapunov exponents,
and fractal dimensions is leavened with creative analogies and many helpful and visually attractive figures and
diagrams. Even more mathematically sophisticated readers will find this book a good starting point in
exploring the complex and beguiling realms of chaos and
f
rada
Is.
Robert C. Hilborn, Associate Executive Officer, American Association of Physics Teachers
ALSO PUBLISHED BY OXFORD UNIVERSITY PRESS:
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Dmitry Budker, Derek Kimball, and David
DeMille
Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers
Robert
С
Hilborn
Chaos and Time-Series Analysis
Julien
Clinton Sprott
|
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| id | DE-604.BV040423022 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:23:44Z |
| institution | BVB |
| isbn | 9780199566440 9780199566433 |
| language | English |
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| spelling | Feldman, David P. Verfasser (DE-588)1027041051 aut Chaos and fractals an elementary introduction David P. Feldman 1. ed. Oxford Oxford Univ. Press 2012 XXI, 408 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chaotisches System (DE-588)4316104-2 gnd rswk-swf Fraktal (DE-588)4123220-3 gnd rswk-swf Fraktal (DE-588)4123220-3 s DE-604 Chaotisches System (DE-588)4316104-2 s Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025275683&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025275683&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Feldman, David P. Chaos and fractals an elementary introduction Chaotisches System (DE-588)4316104-2 gnd Fraktal (DE-588)4123220-3 gnd |
| subject_GND | (DE-588)4316104-2 (DE-588)4123220-3 |
| title | Chaos and fractals an elementary introduction |
| title_auth | Chaos and fractals an elementary introduction |
| title_exact_search | Chaos and fractals an elementary introduction |
| title_full | Chaos and fractals an elementary introduction David P. Feldman |
| title_fullStr | Chaos and fractals an elementary introduction David P. Feldman |
| title_full_unstemmed | Chaos and fractals an elementary introduction David P. Feldman |
| title_short | Chaos and fractals |
| title_sort | chaos and fractals an elementary introduction |
| title_sub | an elementary introduction |
| topic | Chaotisches System (DE-588)4316104-2 gnd Fraktal (DE-588)4123220-3 gnd |
| topic_facet | Chaotisches System Fraktal |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025275683&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025275683&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT feldmandavidp chaosandfractalsanelementaryintroduction |