Chaos and Fractals
Developed by Dale Winter
The mathematics of chaos has received a great deal of public attention (for an area of quite abstract mathematics, anyway)! Most people have seen the beautiful and mysterious pictures produced by chaotic dynamical systems.
The Mandelbrot set:
Click for larger imageThe Julia Set:
Click for larger imageThese are examples of such complicated and beautiful structures. We will try to understand the mathematics behind these incredible images.
Littered across the web pages are opportunities for you to think deeply about the mathematics described. Some of the questions on the web site may seem to be very ambiguous. The questions are intended to be very open-ended, so that you have a lot of freedom to create math for yourself.
The opportunities are indicated with banners. When you see. . .
. . . feel free to pause and think about the material you have just been studying. Often there is more to the material than first meets the eye. You are welcomed and encouraged to talk with your colleagues.
. . . feel free to use a calculator or computer to investigate further aspects of the mathematics that you have been studying. Again, you are very welcome to talk about your investigations with others.
. . . look for links to other web sites that treat similar material.
A Java compatible browser is recommended, as many of the chaos- and fractal-related
web sites feature interactive Java applets that allow you to investigate
the math over the web.
Content of the Course
The subject matter of this course is the mathematical study of dynamical systems. Dynamical Systems is a branch of mathematics that attempts to describe and understand processes in motion.
Many cultures have devised ways of explaining the processes that they observed in nature, and in their everyday lives. For example, many cultures developed elaborate stories to describe and explain the rise and fall, and waxing and waning of the moon.
Do you know of any of the attempts to describe and understand the observed behavior of the moon that other cultures have developed over the millennia?
From the Western-scientific point of view, how would you account for
the observed behavior of the moon? Can you illustrate your arguments with
diagrams to show how your system of explanation describes the behavior
of the moon?
Some dynamical systems, like the phases of the moon, or the tides, are predictable. If you are told the initial set-up of the dynamical system, then you can predict how it will behave with great accuracy.
The motion of the planets around is basically a predictable dynamical system. If you know what the positions of the planets are right now, then by doing some mathematical calculations, you can predict with great accuracy where the planets will be in ten, one hundred or a thousand years from now.
Chaotic dynamical systems are dynamical systems where you just can't make any long-term predictions with any accuracy. The atmosphere is sometimes given as an example of a chaotic dynamical system. Meteorologists can usually make short-term predictions of what the weather will be like, but their long-term predictions almost never turn out to be true.
We will try to develop an understanding of how chaotic dynamical systems differ from other dynamical systems, why they behave differently.
Perhaps surprisingly, you don't have to have very complicated mathematics
to observe and study the apparently complicated subject of chaotic dynamics.
We begin with a description of the process of iterating functions: