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.
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.
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.
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.