Chaos and Fractals: New Frontiers of Science, Second Edition

Cover......Page 1
Chaos and Fractals: New Frontiers of Science (Second Edition)......Page 4
Copyright......Page 5
Preface......Page 6
Preface of the First Edition......Page 7
Authors......Page 11
Contents......Page 12
Foreword......Page 16
Introduction: Causality Principle, Deterministic Laws and Chaos......Page 24
1. The Backbone of Fractals: Feedback and the Iterator......Page 30
1.1 The Principle of Feedback......Page 32
1.2 The Multiple Reduction Copy Machine......Page 38
1.3 Basic Types of Feedback Processes......Page 42
1.4 The Parable of the Parabola — Or: Don’t Trust Your Computer......Page 52
1.5 Chaos Wipes Out Every Computer......Page 64
2. Classical Fractals and Self-Similarity......Page 76
2.1 The Cantor Set......Page 80
2.2 The Sierpinski Gasket and Carpet......Page 91
2.3 The Pascal Triangle......Page 95
2.4 The Koch Curve......Page 102
2.5 Space-Filling Curves......Page 107
2.6 Fractals and the Problem of Dimension......Page 119
2.7 The Universality of the Sierpinski Carpet......Page 125
2.8 Julia Sets......Page 135
2.9 Pythagorean Trees......Page 139
3. Lim and Self-Similarity......Page 144
3.1 Similarity and Scaling......Page 155
3.2 Geometric Series and the Koch Curve......Page 164
3.3 Corner the New from Several Sides: Pi and the Square Root of Two......Page 170
3.4 Fractals as Solutions of Equations......Page 185
4. Length, Area and Dimension: Measuring Complexity and Scaling Properties......Page 196
4.1 Finite and Infinite Length of Spirals......Page 198
4.2 Measuring Fractal Curves and Power Laws......Page 205
4.3 Fractal Dimension......Page 215
4.4 The Box-Counting Dimension......Page 225
4.5 Borderline Fractals: Devil’s Staircase and Peano Curve......Page 233
5. Encoding Images by Simple Transformations......Page 238
5.1 The Multiple Reduction Copy Machine Metaphor......Page 240
5.2 Composing Simple Transformations......Page 243
5.3 Relatives of the Sierpinski Gasket......Page 253
5.4 Classical Fractals by IFSs......Page 261
5.5 Image Encoding by IFSs......Page 267
5.6 Foundation of IFS: The Contraction Mapping Principle......Page 271
5.7 Choosing the Right Metric......Page 281
5.8 Composing Self-Similar Images......Page 285
5.9 Breaking Self-Similarity and Self-Affinity: Networking with MRCMs......Page 290
6. The Chaos Game: How Randomness Creates Deterministic Shapes......Page 300
6.1 The Fortune Wheel Reduction Copy Machine......Page 303
6.2 Addresses: Analysis of the Chaos Game......Page 310
6.3 Tuning the Fortune Wheel......Page 323
6.4 Random Number Generator Pitfall......Page 334
6.5 Adaptive Cut Methods......Page 342
7. Recursive Structures: Growing Fractals and Plants......Page 352
7.1 L-Systems: A Language for Modeling Growth......Page 356
7.2 Growing Classical Fractals with MRCMs......Page 363
7.3 Turtle Graphics: Graphical Interpretation of L-Systems......Page 374
7.4 Growing Classical Fractals with L-Systems......Page 378
7.5 Growing Fractals with Networked MRCMs......Page 390
7.6 L-System Trees and Bushes......Page 395
8. Pascal’s Triangle: Cellular Automata and Attractors......Page 400
8.1 Cellular Automata......Page 405
8.2 Binomial Coefficients and Divisibility......Page 416
8.3 IFS: From Local Divisibility to Global Geometry......Page 427
8.4 HIFS and Divisibility by Prime Powers......Page 435
8.5 Catalytic Converters, or How Many Cells Are Black?......Page 450
9. Irregular Shapes: Randomness in Fractal Constructions......Page 453
9.1 Randomizing Deterministic Fractals......Page 455
9.2 Percolation: Fractals and Fires in Random Forests......Page 459
9.3 Random Fractals in a Laboratory Experiment......Page 470
9.4 Simulation of Brownian Motion......Page 476
9.5 Scaling Laws and Fractional Brownian Motion......Page 486
9.6 Fractal Landscapes......Page 492
10. Deterministic Chaos: Sensitivity, Mixing, and Periodic Points......Page 497
10.1 The Signs of Chaos: Sensitivity......Page 499
10.2 The Signs of Chaos: Mixing and Periodic Points......Page 510
10.3 Ergodic Orbits and Histograms......Page 515
10.4 Metaphor of Chaos: The Kneading of Dough......Page 526
10.5 Analysis of Chaos: Sensitivity, Mixing, and Periodic Points......Page 539
10.6 Chaos for the Quadratic Iterator......Page 550
10.7 Mixing and Dense Periodic Points Imply Sensitivity......Page 559
10.8 Numerics of Chaos: Worth the Trouble or Not?......Page 565
11. Order and Chaos: Period-Doubling and Its Chaotic Mirror......Page 571
11.1 The First Step from Order to Chaos: Stable Fixed Points......Page 578
11.2 The Next Step from Order to Chaos: The Period-Doubling Scenario......Page 589
11.3 The Feigenbaum Point: Entrance to Chaos......Page 605
11.4 From Chaos to Order: A Mirror Image......Page 613
11.5 Intermittency and Crises: The Backdoors to Chaos......Page 625
12. Strange Attractors: The Locus of Chaos......Page 635
12.1 A Discrete Dynamical System in Two Dimensions: Hénon’s Attractor......Page 639
12.2 Continuous Dynamical Systems: Differential Equations......Page 658
12.3 The Rössler Attractor......Page 666
12.4 The Lorenz Attractor......Page 677
12.5 Quantitative Characterization of Strange Chaotic Attractors: Ljapunov Exponents......Page 689
12.6 Quantitative Characterization of Strange Chaotic Attractors: Dimensions......Page 701
12.7 The Reconstruction of Strange Attractors......Page 724
12.8 Fractal Basin Boundaries......Page 736
13. Julia Sets: Fractal Basin Boundaries......Page 745
13.1 Julia Sets as Basin Boundaries......Page 747
13.2 Complex Numbers — A Short Introduction......Page 752
13.3 Complex Square Roots and Quadratic Equations......Page 759
13.4 Prisoners versus Escapees......Page 770
13.5 Equipotentials and Field Lines for Julia Sets......Page 781
13.6 Binary Decomposition, Field Lines and Dynamics......Page 793
13.7 Chaos Game and Self-Similarity for Julia Sets......Page 801
13.8 The Critical Point and Julia Sets as Cantor Sets......Page 806
13.9 Quaternion Julia Sets......Page 817
14. The Mandelbrot Set: Ordering the Julia Sets......Page 820
14.1 From the Structural Dichotomy to the Binary Decomposition......Page 822
14.2 The Mandelbrot Set — A Road Map for Julia Sets......Page 834
14.3 The Mandelbrot Set as a Table of Content......Page 857
1. Books......Page 876
2. General Articles......Page 879
3. Research Articles......Page 880
Index......Page 890