Fractals and Chaos,9780750303996

Fractals and Chaos

by
ISBN13: 9780750303996
ISBN10: 0750303999
Format: Hardcover
Pub. Date: 1997-12-01
Publisher(s): Inst of Physics Pub Inc
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Summary

The aim of this textbook is to provide the reader with an elementary introduction to fractal geometry and chaotic dynamics: subjects which have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. Plenty of problems are provided to test understanding. In addition, advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas.

Table of Contents

Preface xi
1 Introduction
1(7)
1.1 Introduction
1(1)
1.2 A matter of fractals
1(4)
1.3 Deterministic chaos
5(2)
1.4 Chapter summary and further reading
7(1)
1.4.1 Chapter keywords and key phrases
7(1)
1.4.2 Further reading
7(1)
2 Regular fractals and self-similarity
8(19)
2.1 Introduction
8(1)
2.2 The Cantor set
8(2)
2.3 Non-fractal dimensions: the Euclidean and topological dimensions
10(4)
2.4 The similarity dimension
14(2)
2.5 The Koch curve
16(3)
2.6 The quadratic Koch curve
19(1)
2.7 The Koch island
19(2)
2.8 Curves in the plane with similarity dimension exceeding two
21(2)
2.9 The Sierpinski gasket and carpet
23(1)
2.10 The Menger sponge
23(1)
2.11 Chapter summary and further reading
24(1)
2.11.1 Chapter keywords and key phrases
24(1)
2.11.2 Summary and further reading
24(1)
2.12 Revision questions and further tasks
24(3)
3 Random fractals
27(27)
3.1 Introduction
27(1)
3.2 Randomizing the Cantor set and Koch curve
27(2)
3.3 Fractal boundaries
29(1)
3.4 The box counting dimension and the Hausdorff dimension
30(6)
3.5 The structured walk technique and the divider dimension
36(7)
3.6 The perimeter-area relationship
43(1)
3.7 Chapter summary and further reading
43(4)
3.7.1 Chapter keywords and key phrases
43(1)
3.7.2 General
44(1)
3.7.3 Miscellaneous subject areas
44(2)
3.7.4 Perimeter-area relationship
46(1)
3.7.5 Erroneous dimension estimates
46(1)
3.8 Revision questions and further tasks
47(7)
4 Regular and fractional Brownian motion
54(33)
4.1 Introduction
54(1)
4.2 Regular Brownian motion
54(9)
4.3 Fractional Brownian motion: time traces
63(8)
4.4 Fractional Brownian surfaces
71(3)
4.5 Fractional Brownian motion: spatial trajectories
74(5)
4.6 Diffusion limited aggregation
79(1)
4.7 The colour and power of noise
80(1)
4.8 Chapter summary and further reading
81(3)
4.8.1 Chapter keywords and key phrases
81(1)
4.8.2 General
81(1)
4.8.3 Diffusion
82(1)
4.8.4 fBm landscapes etc
83(1)
4.8.5 DLA
83(1)
4.8.6 Power spectra
84(1)
4.9 Revision questions and further tasks
84(3)
5 Iterative feedback processes and chaos
87(30)
5.1 Introduction
87(1)
5.2 Population growth and the Verhulst model
87(1)
5.3 The logistic map
88(1)
5.4 The effect of variation in the control parameter
89(4)
5.4.1 a = 0.90
89(2)
5.4.2 a = 2.60
91(1)
5.4.3 a = 3.20
91(1)
5.4.4 a = 3.52
92(1)
5.4.5 a = 4.00
92(1)
5.5 The general form of the iterated solutions of the logistic map
93(1)
5.6 Graphical iteration of the logistic map
93(5)
5.7 Bifurcation, stability and the Feigenbaum number
98(4)
5.8 A two-dimensional map: the Henon model
102(4)
5.9 Iterations in the complex plane: Julia sets and the Mandelbrot set
106(5)
5.10 Chapter summary and further reading
111(2)
5.10.1 Chapter keywords and key phrases
111(1)
5.10.2 General
111(1)
5.10.3 Maps
111(1)
5.10.4 The Mandelbrot set
112(1)
5.10.5 Iterated function systems
113(1)
5.11 Revision questions and further tasks
113(4)
6 Chaotic oscillations
117(38)
6.1 Introduction
117(1)
6.2 A simple nonlinear mechanical oscillator: the Duffing oscillator
117(8)
6.3 Chaos in the weather: the Lorenz model
125(7)
6.4 The Rossler systems
132(3)
6.5 Phase space, dimension and attractor form
135(3)
6.6 Spatially extended systems: coupled oscillators
138(3)
6.7 Spatially extended systems: fluids
141(4)
6.7.1 Taylor-Couette flow
143(2)
6.8 Mathematical routes to chaos and turbulence
145(1)
6.9 Chapter summary and further reading
146(7)
6.9.1 Chapter keywords and key phrases
146(1)
6.9.2 General
147(1)
6.9.3 The Duffing oscillator
147(1)
6.9.4 The Lorenz system
148(1)
6.9.5 The Rossler systems
148(1)
6.9.6 Fluids and other spatially extended systems
149(1)
6.9.7 Miscellaneous subject areas
150(3)
6.9.8 Numerical methods
153(1)
6.10 Revision questions and further tasks
153(2)
7 Characterizing chaos
155(36)
7.1 Introduction
155(1)
7.2 Preliminary characterization: visual inspection
155(1)
7.3 Preliminary characterization: frequency spectra
156(3)
7.4 Characterizing chaos: Lyapunov exponents
159(4)
7.5 Characterizing chaos: dimension estimates
163(7)
7.5.1 Box counting dimension
163(1)
7.5.2 The information dimension
163(1)
7.5.3 Correlation dimension
164(3)
7.5.4 The pointwise and averaged pointwise dimension
167(1)
7.5.5 The Lyapunov dimension
168(2)
7.6 Attractor reconstruction
170(6)
7.6.1 Method 1 visual inspection of reconstructed attractors
172(1)
7.6.2 Method 2 dominant period relationship
172(1)
7.6.3 Method 3 the autocorrelation function
172(1)
7.6.4 Method 4 the minimum mutual information criterion
172(4)
7.7 The embedding dimension for attractor reconstruction
176(1)
7.8 The effect of noise
177(2)
7.9 Regions of behaviour on the attractor and characterization limitations
179(5)
7.10 Chapter summary and further reading
184(4)
7.10.1 Chapter keywords and key phrases
184(1)
7.10.2 General
184(1)
7.10.3 Fourier spectra
184(1)
7.10.4 Dimension
184(1)
7.10.5 Lyapunov exponents
185(1)
7.10.6 Embedding
186(1)
7.10.7 Delay reconstruction
186(1)
7.10.8 Noise
186(1)
7.10.9 Limits
187(1)
7.10.10 Advanced topics
187(1)
7.11 Revision questions and further tasks
188(3)
A Computer program for Lorenz equations
191(4)
A.1 The Rossler modification
194(1)
B Experimental chaos
195(9)
B.1 The buckled beam
195(1)
B.2 The journal bearing
195(3)
B.3 Chua's circuit
198(1)
B.4 The Belousov-Zhabotinsky chemical reaction
198(1)
B.5 Taylor-Couette flow
198(3)
B.6 Rayleigh-Benard convection
201(1)
B.7 The ring cavity
202(2)
C Solutions
204(24)
C.1 Solutions to chapter 2
204(1)
C.2 Solutions to chapter 3
205(8)
C.3 Solutions to chapter 4
213(4)
C.4 Solutions to chapter 5
217(7)
C.5 Solutions to chapter 6
224(1)
C.6 Solutions to chapter 7
225(3)
References 228(25)
Index 253

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