24 May 2005

Actually Carrying Through On The Name

Right, so I guess I'll write about chaos a bit here after all... it's really quite an interesting topic.

From m-w.com, chaos is "a state of things in which chance is supreme," "a state of utter confusion," or "a confused mass or mixture" (among other definitions). However, the mathematical definition of chaos (or perhaps more accurately, chaos theory) is rather specific, referring to systems governed by deterministic, non-linear differential or difference equations that display a strong sensitivity to initial conditions. The phrase 'strong sensitivity to initial conditions' means that selecting an initial value of 20 for some parameter will result in far different behavior than selecting 20.00001. Weather is a prime example of this phenomenon, explaining why the weatherman is wrong most of the time and why forecasts are never attempted more than a week or so ahead of time. In order to exactly predict weather patterns, atmospheric temperature, pressure, humidity, wind speed, wind direction, etc. would need to be known effectively to an infinite number of decimal places. Not so easy.

One curious (and frustrating - I will likely face this in my research) problem with the numerical simulation of chaotic systems is that by their nature, computers themselves can only handle a finite number of decimal places. Thus, even if you were (somehow!) able to exactly match the real starting conditions to those in your simulation, after the first iteration in the computer rounding would occur, causing the simulation to deviate from the real system. Tricky, no? :-)

So, enough of my long-winded description. A picture is worth a thousand words, so here:

Chaos Theory at Wikipedia (Okay, more words, but lots of background and very nifty links!)

Double-pendulum applet - fun to mess around with

Fractal fern, from here - Fractals obey 'self-similarity on multiple scales,' and while not strictly chaotic themselves, some chaotic systems can exhibit fractal-like behavior (again, see Wikipedia for more info).

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