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E-grāmata: Chaos and Fractals: An Elementary Introduction [Oxford Scholarship Online E-books]

(Department od Physics and Mathematics, College of the Atlantic, Bar Harbor, Maine, USA)
  • Formāts: 432 pages, 306 b/w line drawings, 31 b/w halftones
  • Izdošanas datums: 09-Aug-2012
  • Izdevniecība: Oxford University Press
  • ISBN-13: 9780199566433
Citas grāmatas par šo tēmu:
  • Oxford Scholarship Online E-books
  • Cena pašlaik nav zināma
Chaos and Fractals: An Elementary IntroductionPalielināt
  • Formāts: 432 pages, 306 b/w line drawings, 31 b/w halftones
  • Izdošanas datums: 09-Aug-2012
  • Izdevniecība: Oxford University Press
  • ISBN-13: 9780199566433
Citas grāmatas par šo tēmu:
Wiley OnlineMore info about Oxford Scholarship Online e-books
Rokasgrāmatas mājas lapa: http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199566433.001.0001/acprof-9780199566433
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. It introduces the key phenomena of chaos - aperiodicity, sensitive dependence on initial conditions, bifurcations - via simple iterated functions. 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 two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.

The book is richly illustrated and includes over 200 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.
I Introducing Discrete Dynamical Systems
1(74)
0 Opening Remarks
3(6)
0.1 Chaos
3(1)
0.2 Fractals
4(1)
0.3 The Character of Chaos and Fractals
5(4)
1 Functions
9(8)
1.1 Functions as Actions
9(1)
1.2 Functions as a Formula
10(1)
1.3 Functions are Deterministic
11(1)
1.4 Functions as Graphs
11(2)
1.5 Functions as Maps
13(4)
Exercises
14(3)
2 Iterating Functions
17(8)
2.1 The Idea of Iteration
17(1)
2.2 Some Vocabulary and Notation
18(1)
2.3 Iterated Function Notation
19(1)
2.4 Algebraic Expressions for Iterated Functions
20(1)
2.5 Why Iteration?
21(4)
Exercises
23(2)
3 Qualitative Dynamics: The Fate of the Orbit
25(8)
3.1 Dynamical Systems
25(1)
3.2 Dynamics of the Squaring Function
26(1)
3.3 The Phase Line
27(1)
3.4 Fixed Points via Algebra
27(2)
3.5 Fixed Points Graphically
29(1)
3.6 Types of Fixed Points
30(3)
Exercises
31(2)
4 Time Series Plots
33(4)
4.1 Examples of Time Series Plots
33(4)
Exercises
35(2)
5 Graphical Iteration
37(8)
5.1 An Initial Example
37(1)
5.2 The Method of Graphical Iteration
38(1)
5.3 Further Examples
39(6)
Exercises
42(3)
6 Iterating Linear Functions
45(8)
6.1 A Series of Examples
45(3)
6.2 Slopes of +1 or -1
48(5)
Exercises
50(3)
7 Population Models
53(14)
7.1 Exponential Growth
53(3)
7.2 Modifying the Exponential Growth Model
56(3)
7.3 The Logistic Equation
59(3)
7.4 A Note on the Importance of Stability
62(2)
7.5 Other r Values
64(3)
Exercises
65(2)
8 Newton, Laplace, and Determinism
67(8)
8.1 Newton and Universal Mechanics
67(2)
8.2 The Enlightenment and Optimism
69(1)
8.3 Causality and Laplace's Demon
70(1)
8.4 Science Today
71(1)
8.5 A Look Ahead
72(3)
II Chaos
75(80)
9 Chaos and the Logistic Equation
77(12)
9.1 Periodic Behavior
77(5)
9.2 Aperiodic Behavior
82(3)
9.3 Chaos Defined
85(1)
9.4 Implications of Aperiodic Behavior
86(3)
Exercises
87(2)
10 The Butterfly Effect
89(16)
10.1 Stable Periodic Behavior
89(1)
10.2 Sensitive Dependence on Initial Conditions
90(3)
10.3 SDIC Defined
93(2)
10.4 Lyapunov Exponents
95(2)
10.5 Stretching and Folding: Ingredients for Chaos
97(2)
10.6 Chaotic Numerics: The Shadowing Lemma
99(6)
Exercises
102(3)
11 The Bifurcation Diagram
105(10)
11.1 A Collection of Final-State Diagrams
105(5)
11.2 Periodic Windows
110(1)
11.3 Bifurcation Diagram Summary
111(4)
Exercises
112(3)
12 Universality
115(16)
12.1 Bifurcation Diagrams for Other Functions
115(3)
12.2 Universality of Period Doubling
118(3)
12.3 Physical Consequences of Universality
121(3)
12.4 Renormalization and Universality
124(4)
12.5 How are Higher-Dimensional Phenomena Universal?
128(3)
Exercises
129(2)
13 Statistical Stability of Chaos
131(10)
13.1 Histograms of Periodic Orbits
131(1)
13.2 Histograms of Chaotic Orbits
132(3)
13.3 Ergodicity
135(3)
13.4 Predictable Unpredictability
138(3)
Exercises
139(2)
14 Determinism, Randomness, and Nonlinearity
141(14)
14.1 Symbolic Dynamics
141(2)
14.2 Chaotic Systems as Sources of Randomness
143(1)
14.3 Randomness?
144(4)
14.4 Linearity, Nonlinearity, and Reductionism
148(4)
14.5 Summary and a Look Ahead
152(3)
Exercises
154(1)
III Fractals
155(80)
15 Introducing Fractals
157(6)
15.1 Shapes
157(1)
15.2 Self-Similarity
158(2)
15.3 Typical Size?
160(1)
15.4 Mathematical vs. Real Fractals
161(2)
Exercises
162(1)
16 Dimensions
163(10)
16.1 How Many Little Things Fit inside a Big Thing?
163(2)
16.2 The Dimension of the Snowflake
165(1)
16.3 What does D 1.46497 Mean?
166(1)
16.4 The Dimension of the Cantor Set
167(1)
16.5 The Dimension of the Sierpinski Triangle
168(1)
16.6 Fractals, Defined Again
169(4)
Exercises
170(3)
17 Random Fractals
173(14)
17.1 The Random Koch Curve
173(3)
17.2 Irregular Fractals
176(2)
17.3 Fractal Landscapes
178(1)
17.4 The Chaos Game
178(3)
17.5 The Role of Randomness
181(1)
17.6 The Collage Theorem
181(6)
Exercises
184(3)
18 The Box-Counting Dimension
187(8)
18.1 Covering a Box with Little Boxes
187(2)
18.2 Covering a Circle with Little Boxes
189(1)
18.3 Estimating the Box-Counting Dimension
190(3)
18.4 Summary
193(2)
Exercises
193(2)
19 When do Averages Exist?
195(12)
19.1 Tossing a Coin
195(3)
19.2 St. Petersburg Game
198(4)
19.3 Average Winnings for the St. Petersburg Game
202(1)
19.4 Implications
203(4)
Exercises
204(3)
20 Power Laws and Long Tails
207(14)
20.1 The Central Limit Theorem and Normal Distributions
207(4)
20.2 Power Laws: An Initial Example
211(2)
20.3 Power Laws and the Long Tail
213(2)
20.4 Power Laws and Fractals
215(3)
20.5 Where do Power Laws Come From?
218(3)
Exercises
219(2)
21 Infinities, Big and Small
221(14)
21.1 What is the Size of the Cantor Set?
221(1)
21.2 Cardinality, Counting, and the Size of Sets
222(2)
21.3 Countable Infinities
224(1)
21.4 Rational and Irrational Numbers
225(1)
21.5 Binary
225(2)
21.6 The Cardinality of the Unit Interval
227(2)
21.7 The Cardinality of the Cantor Set
229(3)
21.8 Summary and a Look Ahead
232(3)
Exercises
232(3)
IV Julia Sets and the Mandelbrot Set
235(36)
22 Introducing Julia Sets
237(4)
22.1 The Squaring Function
237(1)
22.2 Other Examples
238(1)
22.3 Summary
239(2)
Exercises
240(1)
23 Complex Numbers
241(8)
23.1 The Square Root of -1
241(1)
23.2 The Algebra of Complex Numbers
242(1)
23.3 The Geometry of Complex Numbers
243(1)
23.4 The Geometry of Multiplication
244(5)
Exercises
246(3)
24 Julia Sets for the Quadratic Family
249(8)
24.1 The Complex Squaring Function
249(1)
24.2 Another Example: f(z) = z2 - 1
250(2)
24.3 Julia Sets for f(z) = z2 + c
252(1)
24.4 Computing and Coloring Julia Sets
253(4)
Exercises
255(2)
25 The Mandelbrot Set
257(14)
25.1 Cataloging Julia Sets
257(1)
25.2 The Mandelbrot Set Defined
258(1)
25.3 The Mandelbrot Set and the Critical Orbit
259(1)
25.4 Exploring the Mandelbrot Set
260(3)
25.5 The Mandelbrot Set is a Julia Set Encyclopedia
263(5)
25.6 Conclusion
268(3)
Exercises
269(2)
V Higher-Dimensional Systems
271(80)
26 Two-Dimensional Discrete Dynamical Systems
273(14)
26.1 Review of One-Dimensional Discrete Dynamics
273(1)
26.2 Two-Dimensional Discrete Dynamical Systems
274(1)
26.3 The Henon Map
275(2)
26.4 Chaotic Behavior and the Henon Map
277(2)
26.5 A Chaotic Attractor
279(4)
26.6 Strange Attractors Defined
283(4)
Exercises
285(2)
27 Cellular Automata
287(16)
27.1 One-Dimensional Cellular Automata: An Initial Example
287(3)
27.2 Surveying One-Dimensional Cellular Automata
290(3)
27.3 Classifying and Characterizing CA Behavior
293(3)
27.4 Behavior of CAs Using a Single-Cell Seed
296(2)
27.5 CA Naming Conventions
298(2)
27.6 Other Types of CAs
300(3)
Exercises
302(1)
28 Introduction to Differential Equations
303(10)
28.1 Continuous Change
303(1)
28.2 Instantaneous Rates of Change
304(2)
28.3 Approximately Solving a Differential Equation
306(4)
28.4 Euler's Method
310(1)
28.5 Other Solution Methods
311(2)
Exercises
312(1)
29 One-Dimensional Differential Equations
313(8)
29.1 The Continuous Logistic Equation
313(3)
29.2 Another Example
316(1)
29.3 Overview of One-Dimensional Differential Equations
317(4)
Exercises
318(3)
30 Two-Dimensional Differential Equations
321(14)
30.1 Introducing the Lotka-Volterra Model
321(2)
30.2 Euler's Method in Two Dimensions
323(2)
30.3 Analyzing the Lotka-Volterra Model
325(1)
30.4 Phase Space and Phase Portraits
326(3)
30.5 Another Example: An Attracting Fixed Point
329(1)
30.6 One More Example: Limit Cycles
330(1)
30.7 Overview of Two-Dimensional Differential Equations
331(4)
Exercises
332(3)
31 Chaotic Differential Equations and Strange Attractors
335(16)
31.1 The Lorenz Equations
335(1)
31.2 A Fixed Point
336(2)
31.3 Periodic Behavior
338(1)
31.4 Chaos and the Lorenz Equations
339(3)
31.5 The Lorenz Attractor
342(3)
31.6 The Rossler Attractor
345(2)
31.7 Chaotic Flows and One-Dimensional Functions
347(4)
Exercises
349(2)
VI Conclusion
351(10)
32 Conclusion
353(8)
32.1 Summary
353(1)
32.2 Order and Disorder
354(1)
32.3 Prediction and Understanding
355(1)
32.4 A Theory of Forms
355(2)
32.5 Revolution or Reconfiguration?
357(4)
VII Appendices
361(42)
A Review of Selected Topics from Algebra
363(14)
A.1 Exponents
363(3)
A.2 The Quadratic Formula
366(2)
A.3 Linear Functions
368(2)
A.4 Logarithms
370(4)
Exercises
374(3)
B Histograms and Distributions
377(12)
B.1 Representing Data with Histograms
377(1)
B.2 Choosing Bin Sizes
378(3)
B.3 Normalizing Histograms
381(2)
B.4 Approximating Histograms with Functions
383(3)
Exercises
386(3)
C Suggestions for Further Reading
389(14)
C.1 (Mostly) Books
389(4)
C.2 Peer-Reviewed Papers
393(2)
C.3 Suggestions for Further Reading
395(2)
References
397(6)
Index 403
David Feldman joined the faculty at College of the Atlantic in 1998, having completed a PhD in Physics at the University of California. He served as Associate Dean for Academic Affairs from 2003 - 2007. At COA Feldman has taught over twenty different courses in physics, mathematics, and computer science.

Feldman's research interests lie in the fields of statistical mechanics and nonlinear dynamics. In his research, he uses both analytic and computational techniques. Feldman has authored research papers in journals including Physical Review E, Chaos, and Advances in Complex Systems. In 2011-12 he was a U.S. Fulbright Lecturer in Kigali, Rwanda.
Permanent link: https://www.kriso.lv/db/9780199566433_pe.html
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