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 Chaos

The Science of Discworld talks about Langton's Ant. The idea is a tiny ant is sitting in the middle of a grid of squares that are either black or white. There are three rules:

• During each round, the ant moves forward onto the next square.
• When the ant steps on a square, it changes color from black to white or vice versa.
• When the ant steps on a white square, it turns right. When the
The point of systems such as Langton's Ant is that very simple rules can lead to complex behavior. This is one of the more interesting results of chaos theory. Unfortunately this idea leads to several misconceptions by those who don't understand the topic very well.

First, some people think it means "you cannot understand any complex system" or "all complex systems tend towards chaos." That's what the silly mathematician thought in "Jurassic Park." That's almost the opposite of what Langton's Ant shows.

Langton's Ant shows that simple rules can lead to complex systems. It says nothing about whether complex systems might have simple rules or whether they are stable. Langton's Ant is an apparently complex system with simple rules. A space shuttle is a complex system without simple rules. It contains more than 1 million parts that all need to work more or less correctly to keep the system running smoothly.

I've worked on several software projects that contained many thousands of lines of code and they were quite stable. I've also seen software systems including more than 5 million lines of code that were, if not friendly, at least stable. Finally, a modern television, automobile, or aircraft is extremely complex. The latest stealth aircraft are so complex that a human cannot even keep them in the air without computer assistance. If you don't think these systems are stable, drive down to your local electronics store and buy a new television.

Second, some people think you need to study a chaotic system very carefully to understand what it does. If the system is chaotic, you just need to examine it more carefully to predict its behavior.

On the other hand, some people think you can never make meaningful predictions about a chaotic system no matter how much you study it.

Naturally both positions are right and both are wrong. By definition, you cannot predict the behavior of a chaotic system, at least far in the future.

That doesn't mean you cannot make any predictions at all, just that you cannot get the details right for very long. For example, the weather is a classic chaotic system. I cannot predict with any certainty whether it will snow on this date next year. However, knowing today's weather the weather guesser on my local news station can say with confidence that tomorrow will not bring me rain, thunder showers, hail, or a tornado. It is very likely to be sunny and cold.

Long range forecasters can also predict with some accurracy whether next year's rainfall will be unexpectedly large or small.

Third, because small changes in a chaotic system can lead to big changes in the result, many people believe they always do. The classic example here is that a butterfly sneezes in one part of the world and causes a hurricane in another part of the world. The idea is the small perturbation gets amplified until it causes dramatic results.

Small changes in the initial conditions may lead to dramatic changes in the result but they may not. The outcome depends on how important the change is to the system. Chances are if I sneeze here in Boulder Colorado, the air flutters around for a bit and the surroundung air damps out the disturbance. There is little or no change to the weather system overall. My sneeze will probably not make it rain in London next week (that's going to happen anyway ;-).

On the other hand, some systems are unstable and, whether they are not chaotic or not, a small change can have drastic consequence. When the snowpack is right, stepping on it can cause an avalanche. Removing the bottom level from a house of cards will bring the whole structure down. A few poorly designed ballots leads to to a president who says "subliminable."

Sometimes small changes lead to big differences in the final outcome whether the system is chaotic or not.