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Encounters with Chaos and Fractals

Table of Contents

Periodic Points
Iterates of Functions
Fixed Points
Periodic Points
Families of Functions
The Quadratic Family
Bifurcations
Period-3 Points
The Schwarzian Derivative

One-Dimensional Chaos
Chaos
Transitivity and Strong Chaos
Conjugacy
Cantor Sets

Two-Dimensional Chaos
Review of Matrices
Dynamics of Linear Functions
Nonlinear Maps
The Hénon Map
The Horseshoe Map

Systems of Differential Equations
Review of Systems of Differential Equations
Almost Linearity
The Pendulum
The Lorenz System

Introduction to Fractals
Self-Similarity
The Sierpiński Gasket and Other "Monsters"
Space-Filling Curves
Similarity and Capacity Dimensions
Lyapunov Dimension
Calculating Fractal Dimensions of Objects

Creating Fractals Sets
Metric Spaces
The Hausdorff Metric
Contractions and Affine Functions
Iterated Function Systems
Algorithms for Drawing Fractals

Complex Fractals: Julia Sets and the Mandelbrot Set
Complex Numbers and Functions
Julia Sets
The Mandelbrot Set

Computer Programs

Answers to Selected Exercises

References

Index

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