Chaos

Student: I have played the so-called Chaos Game and experimented with the Simple Forest Fire. Both of these are somehow related to chaos. I'm confused though because I thought chaos was a word to describe disorder such as the chaos of New York traffic.

Mentor: Well, that is one definition of chaos, but the term chaos is also used to express a mathematical theory. You are thinking of chaos in the vernacular sense, but for this activity it is representing a mathematical idea.

Student: OK, I'll try to think of it as a mathematical term. In that case, what is chaos??

Mentor: Let's see if you can figure it out. Chaos can be present in many ways. In the Chaos Game it's present when even though the result of each individual throw of the die is unpredictable there is an overall pattern resulting from many throws of the die. In the Fire!! activity a small change in the beginning probability of burn resulted in different overall percentages of the forest burned.

Student: Oh, I think I'm getting it. Chaos is when there is an overall pattern that results from seemingly random events.

Mentor: Yes, you are getting it. But there is a little more than just that. Chaos is also present when the final outcome of a chain of events becomes unpredictable because of a small change in the events leading up to the outcome. This can also be seen in the Fire!! activity. By slowly changing the probability of burn, the resulting overall percent of the forest burned fluctuates drastically.

E. N. Lorenz first stumbled upon this latter idea in 1956 when he was trying to look at modeling the weather using mathematical equations. He found some good equations to work with (they now bear his name) and saw that if he made very small changes to the initial inputs for the weather, he got dramatically different, unpredictable results (outputs). Moreover, the further into the future he tried to predict, the worse it got!

Student: Weather forecasters seem to be able to predict the weather now at least for a little while into the future.

Mentor: That's because they've had over 40 years to find better equations. But they still don't have great predictions past two or three days.

Student: So if chaos is present, we can't do anything with our computer information? If it is not good for predicting, why do it?

Mentor: There are several reasons to look at chaos. Think about the Chaos game. The unpredictability on the small scale (where to move next) results in a predictable fractal shape! Chaos and Fractals are very closely linked. Chaotic things can end up with very interesting -- and predictable -- fractal patterns. The particular individual numbers are not predictable, but the overall pattern can be. With chaotic systems like weather, we just have to be careful about how far into the future we can predict.