Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science

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1 Hei nz-ottopeitgen Hartmut Jürgens Dietmar Sau pe Chaos and Fractals New Frontiers of Science

3 Preface Authors VU X I Foreword 1 Mitchell J. Feigenbaum Introduction: Causality Principle, Deterministic Laws and Chaos 9 1 The Backbone of Fractals: Feedback and the Iterator The Principle of Feedback The Multiple Reduction Copy Machine Basic Types of Feedback Processes The Parable of the Parabola - Or : Don't Trust Your Computer Chaos Wipes Out Every Computer Program of the Chapter: Graphical Iteration 60 2 Classical Fractals and Self-Similarity The Cantor Set The Sierpinski Gasket and Carpet The Pascal Triangle The Koch Curve Space-Filling Curves Fractals and the Problem of Dimension The Universality of the Sierpinski Carpet Julia Sets Pythagorean Trees Program of the Chapter: Sierpinski Gasket by Binary Addresses Limits and Self-Similarity Similarity and Scaling Geometric Series and the Koch Curve Corner the New from Several Sides: Pi and the Square Root of Two Fractals as Solution of Equations Program of the Chapter: The Koch Curve 179

4 4 Length, Area and Dimension : Measuring Complexity and Scaling Properties Finite and Infinite Length of Spirals Measuring Fractal Curves and Power Laws Fractal Dimension The Box-Counting Dimension Borderline Fractals : Devil's Staircase and Peano Curve Program of the Chapter : The Cantor Set and Devil's Staircase Encoding Images by Simple Transformations The Multiple Reduction Copy Machine Metaphor Composing Simple Transformations Relatives of the Sierpinski Gasket Classical Fractals by IFSs Image Encoding by IFSs Foundation of IFS: The Contraction Mapping Principle Choosing the Right Metric Composing Self-Similar Images Breaking Self-Similarity and Self-Affinity or, Networking with MRCMs Program of the Chapter : Iterating the MRCM The Chaos' Game: How Randomness Creates Deterministic Shapes The Fortune Wheel Reduction Copy Machine Addresses: Analysis of the Chaos Game Tuning the Fortune Wheel Random Number Generator Pitfall Adaptive Cut Methods Program of the Chapter: Chaos Game for the Fern Recursive Structures : Growing of Fractals and Plants L-Systems: A Language For Modeling Growth Growing Classical Fractals with MRCMs Turtle Graphics : Graphical Interpretation of L-Systems Growing Classical Fractals with L-Systems Growing Fractals with Networked MRCMs L-System Trees and Bushes 397, 7.7 Program of the Chapter : L-systems Pascal's Triangle : Cellular Automata and Attractors Cellular Automata Binomial Coefficients and Divisibility IFS: From Local Divisibility to Global Geometry HIFS and Divisibility by Prime Powers Catalytic Converters or how many Cells are Black? Program of the Chapter: Cellular Automata 454

5 9 Irregular Shapes : Randomness in Fractal Constructions Randomizing Deterministic Fractals Percolation : Fractals and Fires in Random Forests Random Fractals in a Laboratory Experiment Simulation of Brownian Motion Scaling Laws and Fractional Brownian Motion Fractal Landscapes Program of the Chapter : Random Midpoint Displacement Deterministic Chaos : Sensitivity, Mixing, and Periodic Points The Signs of Chaos : Sensitivity The Signs of Chaos : Mixing and Periodic Points Ergodic Orbits and Histograms Paradigm of Chaos : The Kneading of Dough Analysis of Chaos: Sensitivity, Mixing, and Periodic Points Chaos for the Quadratic Iterator Mixing and Dense Periodic Points Imply Sensitivity Numerics of Chaos : Worth the Trouble or Not? Program of the Chapter : Time Series and Error Development Order and Chaos : Period-Doubling and its Chaotic Mirror The First Step From Order to Chaos : Stable Fixed Points The Next Step From Order to Chaos : The Period Doubling Scenario The Feigenbaum Point: Entrance to Chaos From Chaos to Order : a Mirror Image Intermittency and Crises : The Backdoors to Chaos Program of the Chapter: Final State Diagram Strange Attractors: The Locus of Chaos A Discrete Dynamical System in Two Dimensions : Henon's Attractor Continuous Dynamical Systems : Differential Equations The Rössler Attractor The Lorenz Attractor Quantitative Characterization of Strange Chaotic Attractors : Ljapunov Exponents Quantitative Characterization of Strange Chaotic Attractors : Dimensions The Reconstruction of Strange Attractors Fractal Basin Boundaries 757, 12.9 Program of the Chapter: Rössler Attractor Julia Sets : Fractal Basin Boundaries Julia Sets as Basin Boundaries Complex Numbers - A Short Introduction Complex Square Roots and Quadratic Equations ' Prisoners versus Escapees Equipotentials and Field Lines for Julia Sets Binary Decomposition, Field Lines and Dynamics 812

6 13.7 Chaos Game and Self-Similarity for Julia Sets The Critical Point and Julia Sets as Cantor Sets Quaternion Julia Sets Program of the Chapter : Julia Sets The Mandelbrot Set: Ordering the Julia Sets From the Structural Dichotomy to the Binary Decomposition The Mandelbrot Set - A Road Map for Julia Sets The Mandelbrot Set as a Table of Content Program of the Chapter: The Mandelbrot Set 89 6 A A Discussion of Fractal Image Compression 90 3 Yuval Fisher A.1 Self-Similarity in Images 90 6 A.2 A Special MRCM 90 8 A.3 Encoding Images 91 2 A.4 Ways to Partition Images 914 A.5 Implementation Notes 91 7 B Multifractal Measures 92 1 Carl J. G. Evertsz and Benoit B. Mandelbrot B.1 Introduction 922 B.2 The Binomial and Multinomial Measures 927 B.3 Methods for Estimating the Function f(a) from Data 93 8 B.4 Probabilistic Roots of Multifractals. Role of f (a) in Large Deviation Theory 944 B.5 Some Applications, and Advanced Multifractals 952 Bibliography 955 Index 971

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