This book offers a fun and enriching introduction to chaos theory, fractals and dynamical systems, and on the applications of fractals to computer generated graphics and image compression. Introduction to Chaos, Fractals and Dynamical Systems particularly focuses on natural and human phenomenon that can be modeled as fractals, using simple examples to explain the theory of chaos and how it affects all of us. Then, using straightforward mathematic and intuitive descriptions, computer generated graphics and photographs of natural scenes are used to illustrate the beauty of fractals and their importance in our world. Finally, the concept of Dynamical Systems, that is, time-dependent systems, the foundation of Chaos and Fractal, is introduced. Everyday examples are again used to illustrate concepts, and the importance of understanding how these vital systems affect our lives. Throughout the fascinating history of the evolution of chaos theory, fractals and dynamical systems is presented, along with brief introductions to the scientists, mathematicians and engineers who created this knowledge. Introduction to Chaos, Fractals and Dynamical Systems contains ample mathematical definitions, representations, discussions and exercises, so that this book can be used as primary or secondary source in home schooling environments. The book is suitable for homeschooling as a focused course on the subject matter or as a classroom supplement for a variety of courses at the late junior high or early high-school level. For example, in addition to a standalone course on Chaos, Fractals and Dynamical Systems (or similar title), this book could be used with the following courses: Precalculus Geometry Computer programming (e.g. Rust, C, C++, Python, Java, Pascal) Computer graphics The text can also be used in conjunction with mathematics courses for undergraduates for non-science majors. The book can also be used for informal and lively family study and discussion. For each chapter, exercises and things to do are included. These activities range from simple computational tasks to more elaborate computer projects, related activities, biographical research and writing assignments.
Author(s): Phillip A Laplante, Chris Laplante
Series: Problem Solving In Mathematics And Beyond, 29
Edition: 1
Publisher: WSPC
Year: 2023
Language: English
Commentary: Publisher PDF | Published: August 2023
Pages: 216
City: New Jersey
Tags: Fractals; Dynamics; System Chaotic Behavior
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
