Table of Contents

Preface

1. Mathematical Experiments
   The first chapter introduces the basics of one-dimensional iterated maps. Say what? Take a function y = f(x). Substitute some number into it. Take the answer and run it through the function again. Keep doing this forever. This is called iteration. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. In the 1970's, a whole new branch of mathematics arose from the simple experiments described in this chapter.
   1. Iteration & Orbits
   2. Orbit & Bifurcation Diagrams
   3. Universality
      
      
2. Strange & Complex
   The second chapter extends the idea of an iterated map into two dimensions, three dimensions, and complex numbers. This leads to the creation of mathematical monsters called fractals. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. They are quite interesting to look at and have captured a lot of attention. This chapter describes the methods for constructing some of them.
   1. Strange Attractors
   2. Julia Sets
   3. Mandelbrot Sets
      
      
3. What is Dimension?
   The third chapter deals with some of the definitions and applications of the word dimension. A fractal is an object with a fractional dimension. Well, not exactly, but close enough for now. What does this mean? The answer lies in the many definitions of dimension.
   1. General Dimension
   2. Topological Dimension
   3. Fractal Dimension
      
      
4. Measuring Chaos
   The fourth chapter compares linear and non-linear dynamics. The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. The logistic equation is a discrete, second-order, difference equation used to model animal populations. So similar and yet so alike. The harmonic oscillator is quite well behaved. The paramenters of the system determine what it does. The logistic equation is unruly. It jumps from order to chaos without warning. A parameter that discriminates among these behaviors would enable us to measure chaos.
   1. Harmonic Oscillator
   2. Logistic Equation
   3. Lyapunov Exponent
   4. Lyapunov Space

Appendices

1. Printed Resources
2. Software Resources
3. Internet Resources

	This page is a part of E-World! Maintained by Glenn Elert Last modified Sunday, 05-Apr-1998 13:34:23 EDT Go to the nearest index