Fractals and Chaos: An illustrated courseCRC Press, 1 janv. 1997 - 256 pages Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that 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. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study. |
Table des matières
11 Introduction 12 A matter of fractals | xi |
13 Deterministic chaos | 3 |
142 Further reading | 5 |
Regular fractals and selfsimilarity 21 Introduction 22 The Cantor set | 6 |
the Euclidean and topological dimensions | 8 |
24 The similarity dimension | 12 |
27 The Koch island | 17 |
28 Curves in the plane with similarity dimension exceeding two | 19 |
coupled oscillators | 136 |
fluids | 139 |
671 TaylorCouette flow | 141 |
68 Mathematical routes to chaos and turbulence | 143 |
69 Chapter summary and further reading 691 Chapter keywords and key phrases | 144 |
693 The Duffing oscillator | 145 |
695 The Rossler systems | 146 |
696 Fluids and other spatially extended systems | 147 |
210 The Menger sponge | 21 |
2112 Summary and further reading 212 Revision questions and further tasks | 22 |
31 Introduction 32 Randomizing the Cantor set and Koch curve | 25 |
33 Fractal boundaries | 27 |
34 The box counting dimension and the Hausdorff dimension | 28 |
35 The structured walk technique and the divider dimension | 34 |
37 Chapter summary and further reading 371 Chapter keywords and key phrases | 41 |
372 General | 42 |
374 Perimeterarea relationship | 44 |
38 Revision questions and further tasks | 45 |
Regular and fractional Brownian motion 41 Introduction 42 Regular Brownian motion | 52 |
time traces | 61 |
44 Fractional Brownian surfaces | 69 |
spatial trajectories | 72 |
46 Diffusion limited aggregation | 77 |
47 The colour and power of noise | 78 |
482 General | 79 |
483 Diffusion | 80 |
485 DLA | 81 |
49 Revision questions and further tasks | 82 |
Iterative feedback processes and chaos 51 Introduction 52 Population growth and the Verhulst model | 85 |
53 The logistic map | 86 |
541 a 090 | 87 |
543 a 320 | 89 |
56 Graphical iteration of the logistic map | 91 |
Julia sets and the Mandelbrot set | 104 |
5102 General 5103 Maps | 109 |
5104 The Mandelbrot set | 110 |
511 Revision questions and further tasks | 111 |
61 Introduction 62 A simple nonlinear mechanical oscillator the Duffing oscillator | 115 |
the Lorenz model | 123 |
64 The Rossler systems | 130 |
65 Phase space dimension and attractor form | 133 |
697 Miscellaneous subject areas | 149 |
610 Revision questions and further tasks | 151 |
71 Introduction 72 Preliminary characterization visual inspection | 153 |
frequency spectra | 154 |
Lyapunov exponents | 157 |
dimension estimates 751 Box counting dimension 752 The information dimension | 161 |
753 Correlation dimension | 162 |
754 The pointwise and averaged pointwise dimension | 165 |
755 The Lyapunov dimension | 166 |
76 Attractor reconstruction | 168 |
762 Method 2 dominant period relationship | 170 |
77 The embedding dimension for attractor reconstruction | 174 |
78 The effect of noise | 175 |
79 Regions of behaviour on the attractor and characterization limitations | 177 |
7103 Fourier spectra 7104 Dimension | 182 |
7105 Lyapunov exponents | 183 |
7107 Delay reconstruction 7108 Noise | 184 |
71010 Advanced topics | 185 |
711 Revision questions and further tasks | 186 |
Computer program for Lorenz equations | 189 |
AI The Rossler modification | 192 |
BI The buckled beam B2 The journal bearing | 193 |
B4 The BelousovZhabotinsky chemical reaction B5 TaylorCouette flow | 196 |
B6 RayleighBenard convection | 199 |
B7 The ring cavity | 200 |
C1 Solutions to chapter 2 | 202 |
C2 Solutions to chapter 3 | 203 |
C3 Solutions to chapter 4 | 211 |
C5 Solutions to chapter 6 | 222 |
C6 Solutions to chapter 7 | 223 |
References | 226 |
Index | 251 |
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Expressions et termes fréquents
behaviour bifurcation box counting dimension Cantor set chaotic attractor chaotic dynamics chaotic motion chapter co-ordinate coastline complex map construction contains control parameter correlation dimension delay diffusion dimension estimate divider dimension Duffing oscillator dynamical system embedding dimension Euclidean experimental fBm trace fBm trajectory fixed point fluid fractal curve fractal dimension fractal geometry fractal object fractal structure frequency Gaussian Hénon Hurst exponent hypercubes hypersphere hypervolume images initial conditions Julia set keywords and key Koch curve Lett line segments log(N logistic map Lorenz attractor Lorenz system Lyapunov exponents Mandelbrot set measure method mutual information noise nonlinear orbit particle period doubling periodic attractor phase portraits phase space Phys plane power spectra random fractals reconstructed attractor Reynolds number Rössler system scales self-similar shown in figure side length similarity dimension slope spatial step strange attractor surface topological dimension turbulence Xn+1 zero zoom