Introduction To Chaos, Fractals And Dynamical Systems [1 ed.] 9811273243, 9789811273247, 9789811273902, 9789811273254, 9789811273261

Table of contents :
Contents
About the Authors
Acknowledgments
List of Figures
List of Tables
Introduction
1. Background and Objectives
2. Mathematical Background
3. Rust and OpenGL
3.1 The Rust Program Language
3.2 OpenGL and Hardware-Accelerated Graphics
3.3 OpenGL Coordinate System
3.4 Vertices, Shaders and the OpenGL Render Pipeline
4. Running the Programs
5. Organization
6. About the Images
7. OurWish
1. What is Chaos? What are Fractals?
1.1 Stable/Unstable Systems
1.2 What is Chaos?
1.3 What are Fractals?
1.4 Other Fractal-like Things
1.5 How are Fractals Created?
1.5.1 Dynamical Systems
1.5.2 Algorithms
1.5.3 Attracting and Escaping Points
1.5.4 Bifurcation Diagrams
1.6 The Sierpinski Triangle
1.7 Iterated Function System Transformations
1.8 Recursive Generation of Fractals
1.8.1 The Cantor Set
1.9 Fractal Dimension
1.10 How are Fractals and Chaos Related?
1.11 Brief History of Chaos, Fractals and Dynamical Systems
1.12 Exercises and Things to Do
2. Foundations of Chaos and Fractal Theory
2.1 Complex Numbers and Functions
2.1.1 Plotting Complex Numbers
2.1.2 Arithmetic with Complex Numbers
2.2 Functions of Complex Variables
2.3 Generating Fractals by Finding Attractors of Complex Functions
2.3.1 Julia Sets
2.3.2 The Histogram Coloring Algorithm
2.3.3 More Julia Sets
2.3.4 The Mandelbrot Set
2.3.5 A Note on the Images
2.3.6 The Inverse Iteration and Boundary Scanning Methods
2.4 Three-Dimensional Fractals
2.5 Wavelets
2.6 Exercises and Things to Do
3. Chaos and Fractals in Nature
3.1 Population Dynamics
3.2 Animal Images
3.3 Genetics
3.4 Weather
3.5 Scenes from Nature
3.5.1 Trees, Leaves and Flowers
3.5.2 Clouds
3.5.3 Rocks and Boulders
3.5.4 Snowflakes
3.5.5 Galaxies
3.5.6 Coastlines
3.6 Fractals in the Human Body
3.6.1 Bronchial Growth
3.6.2 Neuron Growth
3.6.3 Physiological Processes
3.6.4 Chaos of the Mind?
3.7 Exercises and Things to Do
4. Chaos and Fractals in Human-Made Phenomena
4.1 Turbulent Flow
4.2 Structures
4.3 Computer Scene Analysis
4.4 Image Compression
4.4.1 Problems with Fractal Compression
4.5 Economic Systems
4.6 Cellular Automata
4.6.1 One-Dimensional Cellular Automata
4.6.2 Two-Dimensional Cellular Automata
4.7 Exercises and Things to Do
5. Dynamical Systems and Systems Theory
5.1 Basic System Theory
5.1.1 Modeling Things as Systems
5.1.2 Linear Growth
5.1.3 Response Time
5.1.4 Control Systems
5.1.5 Deterministic Systems
5.1.6 Time-Variant and Time-Invariant Systems
5.1.7 Linear Systems
5.1.8 Nonlinear Systems
5.1.9 Summary of System Types
5.2 Systems Thinking
5.2.1 Emergent Behavior
5.2.2 Second-Order Effects
5.3 Uncertainty
5.3.1 State Uncertainty
5.3.2 Heisenberg Uncertainty
5.3.3 Uncertainty and Chaos
5.3.4 Dealing with Uncertainty
5.3.4.1 Expert Systems
5.3.4.2 Probabilistic Reasoning
5.3.4.3 Fuzzy Values, Fuzzy Logic and Fuzzy Sets
5.3.4.4 Neural Networks
5.3.4.5 Rough Sets
5.3.4.6 Multiple Sources of Uncertainty
5.4 Swarm Behavior
5.4.1 Simulating a Swarm
5.4.2 Applications of Simulated Swarms
5.4.3 Swarm Computing
5.5 Brief History of Dynamical Systems and Systems Theory
5.5.1 Dynamical Systems
5.5.2 Systems Engineering
5.5.3 Origins of Systems Thinking
5.6 Exercises and Things to Do
Glossary
Bibliography
Index

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