CHAOTIC  DYNAMICS: FRACTALS, TILINGS AND SUBSTITUTIONS.
	Mathematica Labs to accompany the Cambridge University Press 
	textbook of Geoffrey Goodson.

These notes were written using Mathematica 10.3, and will need a recent version of Mathematica to be executed. There is a hyperlink to take you to each chapter (click on the dark blue part of the title). These labs are a work in progress, and will be updated from time to time. Comments are most welcome (please email ggoodson@towson.edu). The chapter numbers correspond to the chapters of the textbook.

Acknowledgments

The following were helpful in the preparation of these notebooks:

1. Mathematica in Action, by Stan Wagon.

2. Chaos, Fractals and Dynamics, by Robert Devaney.

3. Chaos and Fractals, by K. Rypdal and M. Rypdal.

4. An Elementary Introduction to the Wolfram Language, by Stephan Wolfram.

5. Dynamical Systems with Applications Using Mathematica, by Stephen Lynch.
 
I would also like to thank my son Garth for some technical assistance.

CONTENTS

CHAPTER 1(a),  One Dimensional Maps.

1.1 Iterations of Maps.

1.2 Web Plots.

CHAPTER 1(b),  Newton’s Method

1.3 Newton’s Method for One-Dimensional Maps

CHAPTER 2(a),  Periodic Points.

2.1 Periodic Points of Quadratic Maps and the Logistic Maps.

2.2 The Tent Family.

2.3 The Creation of the 2-Cycle for c(x)=a cos(x)  and  s(x)=a sin(x) .

2.4  Finding the Periodic Points of (Subscript[L, μ])(x)=μ x(1-x).

CHAPTER 2(b),  Bifurcation Diagram for the Logistic Map

2.5 Bifurcation Diagram for the Logistic Maps.

2.6 The Curves Appearing in the Bifurcation Diagram.

CHAPTER 3,  Sharkovsky’s Theorem.

3.1 The Converse of Sharkovsky’s Theorem and Tent Maps.

CHAPTER 4, Diffeomorphisms.

4.1 Diffeomorphisms.

4.2 Linear Fractional Transformations.

CHAPTER 5,  Cantor Sets.

5.1 Cantor Sets Arising From Logistic Maps.

5.2 Cantor Sets Arising From Tent Maps.

CHAPTER 6,  Chaos.

6.1 Some Chaotic Maps.

6.2 Sensitive Dependence on Initial Conditions.

6.3 Probability Distributions.

CHAPTER 8,  The Schwarzian Derivative

8.1 The Schwarzian Derivative

CHAPTER 9,  The Tent Map

9.1 The Tent Map

CHAPTER 10, Fractals.

10.1 Fractals.

10.2 The Koch Snowflake.

10.3 Fractal Dimension.

10.4 A Fractal Gasket.

10.5 A Square Gasket.

10.6 The Sierpinski Carpet.

10.7 The Sierpinski Triangle.

10.8 Space Filling Curves.

CHAPTER 11,  Newton's Method for Real Quadratics and Cubics.

11.1 Iteration of Maps of the Form  f(x)=(x+a)((x^2)+b).

11.2 A More Dynamical Iteration Method.

CHAPTER 13,  Linear Transformations and the Henon Map.

13.1 Linear Transformations.

13.2 The Henon Map.

CHAPTER 14(a),  Complex Dynamics.

14.1 Julia Sets and Complex Analytic Dynamics.

14.2 Quadratic Maps.

14.3 Julia Sets Continued.

14.4 The Mandelbrot Set.

14.5 The Main Bulb of the Mandelbrot Set.

CHAPTER 14(b),   Newton’s Method for Complex Cubics.

14.6 Newton’s Method for Cubics.

CHAPTER 15,  Substitutions.

15.1 Substitution Sequences.

15.2 Dragon Curves.

15.3 The Positive Paper Folding Substitution.

15.4 The Morse Substitution.

15.5 A 3-Symbol Version of the Morse Substitution.

15.6 The Rudin-Shapiro Substitution.

15.7 The Fibonacci Substitution.

15.8 Another Version of the Morse Substitution.

CHAPTER 16,  2D Substitutions.

16.1 Substitution Sequences.

16.2 The Rauzy Substitution and Incidence Matrices.

16.3 Rauzy Fractals.

16.4 More Rauzy Fractals.

16.5 Dragon Curves.

CHAPTER 19,  Cutting Sequences.

19.1 Cutting Sequences.

CHAPTER 21,  Interval Exchange Transformations.

21.1 Interval Exchange Transformations.

21.2 Exchanges of Three Intervals.

21.3 Interval Exchange Transformations with a Flip.