Tabla de Contenidos:
- Machine generated contents note: 1. Fractals and Fractal Distributions 1
- 1.1 The Fractal Dimension3
- 1.2 Determining Fractal Dimensions8
- 1.3 Fractal Distributions13
- 1.4 Fractals or Fractal Distributions? 17
- 1.5 Are Fractals Useful?19
- 1.6 Where do Fractals Come From? 21
- 2. Recognizing Power-Law Distributions25
- 2.1 Maximum Likelihood, Least Squares, and Linear Regression 26
- 2.2 Do Cumulative Size Distributions Tell the Truth?28
- 2.3 Binning31
- 2.4 Censoring37
- 3. Self-Affine Time Series41
- 3.1 Brownian Motion42
- 3.2 White Noise44
- 3.3 Fourier Transforms46
- 3.4 Fractional Brownian Motion48
- 3.5 Generating FBM51
- 3.6 Scaling Properties of FBM52
- 3.7 Self-Affine Scale Invariance and Fractal Dimensions55
- 3.8 Recognizing FBM56
- 3.9 The Variogram Analysis59
- 3.10 Predictability64
- 4. Deterministic Chaos67
- 4.1 The Lorenz Equations68
- 4.2 The Physics Behind the Lorenz Equations69
- 4.3 Phase Space, Attractors, and Bifurcations74
- 4.4 Limit Cycles and Strange Attractors77
- 4.5 The Lyapunov Exponent78
- 4.6 Does it Matter whether God Plays Dice?84
- 4.7 Deterministic Chaos and Self-Affine Fractals85
- 5. Self-Organized Criticality87
- 5.1 Critical-Point Phenomena88
- 5.2 The Bak-Tang-Wiesenfeld Model90
- 5.3 The Critical State98
- 5.4 What is SOC?99
- 5.5 Sandpile Dynamics and the BTW Model102
- 6. The Forest-Fire Model - Tuning and Universality109
- 6.1 The Forest-Fire Model109
- 6.2 Universality119
- 6.3 Non-Equilibrium States in SOC Systems122
- 7. Earthquakes and Stick-Slip Motion125
- 7.1 The Fractal Character of Earthquakes127
- 7.2 The Burridge-Knopoff Model130
- 7.3 Separation of Time Scales133
- 7.4 Cellular Automata135
- 7.5 The Olami-Feder-Christensen Model138
- 7.6 Boundary Conditions in the OFC Model142
- 7.7 Efficient Simulation of the OFC Model143
- 7.8 Is the OFC Model Self-Organized Critical?145
- 7.9 Rupture Area and Seismic Moment149
- 7.10 The Temporal Fingerprint of the OFC Model152
- 7.11 How Complex is the OFC Model?161
- 8. Landslides163
- 8.1 Fractal Properties of Landslides164
- 8.2 Are Landslides like Sandpile Avalanches?167
- 8.3 Data, Models, and Reality173
- 8.4 The Role of Time-Dependent Weakening175
- 8.5 On Predicting Slope Stability184
- 8.6 Are SOC and Universality Important in Landform Evolution? 187
- 9. Drainage Networks189
- 9.1 Fractal Properties of Drainage Networks189
- 9.2 Discharge, Drainage Areas, and Water Balance196
- 9.3 Peano's Basin198
- 9.4 Random-Walk Approaches200
- 9.5 Drainage Networks and Landform Evolution201
- 9.6 Optimal Channel Networks216
- 9.7 Drainage Networks and Self-Organized Criticality220
- 9.8 Optimization by Permanent Reorganization?233
- 10. SOC and Nothing Else?235
- 10.1 Ensembles of SOC systems236
- 10.2 SOC in Pre-Structured Systems240
- 10.3 Highly Optimized Tolerance245
- 11. Where do we Stand?253
- A. Numerics of Ordinary Differential Equations255.
