Fractals and Chaos

by Addison, Paul S.

• ISBN13: 9780750303996
• ISBN10: 0750303999
• Format: Hardcover
• Copyright: 1997-12-01
• Publisher: Inst of Physics Pub Inc

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