Introduction to Chaos Theory
I recently took a course on Chaos Theory as part of a summer camp at the
University of Chicago. It was
taught by Greg Huber (g-huber@uchicago.edu). If you have any difficult questions you might want to ask him.
The first thing we did in our chaos theory course
was to look at what happens to an initial value, x0, when a function, f(x),
is repeatedly applied to it. We tried this with three functions, both with
domain and range within [0, 1].
The Shift Map
The formula for the shift map is f(x) =
x*2 for 0 <= x < .5
x*2-1 for .5 <= x < 1
It turns out that this function can be understood by writing x0 as a
"binary decimal". For example .1101 is 13/16, 1/2 + 1/4 + 1/16. Each
iteration of the function is equivalent to shifting the decimal point right
one place, and chopping off any integer part. Therefore, to understand this
function, you just have to write x0 as a "binary decimal".
The Tent Map
The formula for the tent map is f(x) =
x*2 for 0 <= x < .5
x*-2+2 for .5 <= x < 1
This function can also be understood fairly simply. First, express x0 as a
"binary decimal". Each iteration of the function starts by shifting the
decimal point one place to the left. If the number is 0 point something,
you are done. However, If the number is one point something turn all of the
ones in the number to zeros, and vice-versa. Therefore, the tent map can
also be understood by expressing x0 as a binary decimal.
The Logistic Map
The formula for the logistic map is f(x) = r * x * (1 - x), where r is a
constant. The height of the map is r/4, so r ranges from 0 to 4.
behavior of the logistic map depends greatly on the value of r. If r is
between 0 and 1, repeated iteration of the logistic map will eventually
give 0. For r between 1 and 3, the logistic map will eventually settle
down to fixed point. After three, more interesting things start to
happen. For r just greater than 3, the logisitic map will start to bounce
back and forth between two numbers. As r gets a little bigger, the
logisitic map will go between four numbers. The number of points
the logistic map settles down to will keep doubling until r becomes
approximately 3.6. Then the logistic map will end in chaos.
If you make a graph with r as the x-axis going from 0 to 4, and
the value the logistic map settles to as the y-axis, you get what
is known as the bifurcation tree. It looks like this:
The Bifurcation Tree
If you look at the bifurcations of the tree, you can see that they happen
faster and faster, until at a certain point, they happen infinitely fast.
The ratio of the bifurcations of the tree is called Feigenbaum's Delta.
Fractals
Another part of chaos theory deals with fractals. The formal definition of a
fractal is fairly complicated and I don't know what it is. However I do
know a little bit about them. Fractals are basicly self-similar objects.
For example, take a line segment with length 1. Then split it into thirds
and get rid of the middle third. Every time you do this, you end up
with two copies of the original, only they are one third the size. This
fractal is known as the Cantor set. Here is a picture of it:
The Cantor Set
Another example is called the Koch Curve. Start with a line segment of length
1. Then replace the middle third of the segment with two segments of length
one third that form the two legs of a equilateral triangle, with the third
side being the line segment you just got rid of. If you keep doing this, you
get something that looks pretty cool.
The classic example of a fractal is the Mandelbrot set. This is a real cool
looking thing that is created a lot differently than most other fractals.
Chaos Links
- The Julia Set
- This is the source code to a C++ program I wrote. It draws the Julia set, and lets you zoom in on different parts of it.
- The Mandelbrot Set
- This is the source code to a C program I wrote. It draws the Mandelbrot set, and lets you zoom in on different parts of it.
- Fractals and Chaos Theory
- A good introduction to chaos theory, including the Bifurcation Tree, and the Mandelbrot Set.
- Chaos Theory: A Brief Introduction
- Another introduction to chaos theory, with many excerpts from James Gleick's book, Chaos.
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David Mellis / stendahl@imsa.edu / Last Modified August 13, 1996