Ian Malcolm's Chaos Theory

from Ian Malcolm's Chaos Theory Homepage

http://www.geocities.com/CapeCanaveral/6211/

 

What may have driven you here is the word "chaos." Physics has had great success at describing certain kinds of behavior : planets in orbit, spacecraft going to the moon, pendulums, springs, rolling balls, and that sort of thing. The regular movement of objects. These are described by what are called "linear equations," and mathematicians can solve those equations easily. We've been doing it for hundreds of

years.

 

But there is another kind of behavior, which physics handles badly. For example, anything to do with turbulence. Water coming out of a spout. Air moving over an airplane wing. Weather. Blood flowing through the heart. Turbulent events are described by "nonlinear equations." They're hard to solve - in fact, they're usually impossible to solve. So physics has never understood this whole class of events.

Until about ten years ago. The new theory that describes them is called "Chaos Theory."

 

Chaos Theory originally grew out of attempts to make computer models of weather in the 1960's. Weather is a big complicated system, namely the earth's atmosphere as it interacts with the land and the sun. The behavior of this big complicated system always defied understanding. So naturally we couldn't predict weather. But what the early researchers learned from computer models was that even if you could

understand it, you still couldn't predict it. Weather prediction is absolutely impossible. The reason is that the behavior of the system is sensitively dependent on initial conditions.

 

If I use a cannon to fire a shell of a certain weight, at a certain speed, and a certain angle of inclination - and if I then fire a second shell with almost the same weight, speed, and angle - what will happen? The two shells will land at almost the same spot. That's linear dynamics.

 

But if I have a weather system that I start up with a certain temperature and a certain temperature and a certain wind speed and a certain humidity - and if I then repeeaat it with almost the same temperature, wind, and humidity - the second system will not behave the same. It'll wander off and rapidly will become very differant from the first. Thunderstorms instead of sunshine. That's nonlinear dynamics. They are sensitive to initial conditions : tiny differences become amplified.

 

The shorthand is the 'butterfly effect.' A butterfly flaps its wings in Peking, and weather in New York is different.

 

You may think chaos is all just random and unpredictability, but its actually not. We find hidden regularities within the complex variety of a system's behavior. That's why chaos has now become a very broad theory that's used to study everything from the stock market, to rioting crowds, to brain waves during epilepsy. Any sort of complex system where there is confusion and unpredictability. We can find an underlying order, which is characterized by the movement of the system within phase space.

 

Chaos Theory says two things. First, that complex systems like weather have an underlying order. Second, the reverse of that - that simple systems can produce complex behavior. For example, pool balls. You hit a pool ball, and it starts to carom off the sides of the table. In theory, that's a fairly simple system, almost a Newtonian system. Since you can know the force imparted to the ball, and the mass of the ball, and you can calculate the angles at which it will strike the walls, you could predict the behavior of the ball far into the future, as it keeps bouncing from side to side. You could predict where it will end up three hours from now, in theory.

 

But in fact, it turns out you can't predict more than a few seconds into the future. Becaus almost immediately very small effects - imperfections in the surface of the ball, tiny indentations in the wood of the table - start to make a difference. And it doesn't take long before they overpower your careful calculations. So it turns out that this simple system of a pool ball on a table has unpredictable behavior.