shallowdeep
Posts: 343
Joined: 9/1/2006
From: California
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quote:
ORIGNINAL: subtee
Would someone please dumb this down for me, say to a fifth gr---third grade level?
I'm an expert on neither chaos theory, nor third grade educational practices... but, here's an attempt. While I haven't personally read it, the book that thornhappy recommended, Chaos: Making a New Science by James Gleick, is generally well regarded and is most likely a more entertaining and informative read than this post. You've been warned.
Chaos theory comprises the study of certain systems with complex, seemingly random behavior. However, the systems are not random. They are clearly defined by a set of equations (i.e. they are deterministic). What they do lack is an analytical solution - there is no way to find an explicit equation representing the solution. It's a bit like having a recipe with a bunch of steps, but no way of knowing what it's going to make until you try it.
When people first started carefully investigating such systems in the late 1800s, they got far enough to realize that they didn't want to try going much further. Once you realize you can't get an analytical solution (i.e. you realize you can't remember what the recipe was for) you can still use numerical analysis (i.e. follow the recipe step by step) to see what happens. You have the steps, and they aren't even that hard... in some cases a third grader could do them. But there is a problem: to follow the recipe perfectly requires an infinite number of steps. Even getting a good approximation requires so many steps that even the brightest mathematicians with the most boring social lives of the time didn't want to try. As a result, linear systems got most of the attention.
Of course computers are very good at doing a bunch of simple, repetitive calculations very quickly - and they don't have any social lives at all. So, once they became common, it didn't take too long for some progress to be made. Lorenz happened to be using an early computer to do some weather model simulations when he made an interesting observation, which we'll get to in a moment.
As has already been mentioned, a defining characteristic of chaotic systems is that they are very sensitive to initial conditions. All this means is that a very small difference in the way things are set up will have a vary large impact on what happens down the road.
Imagine dropping a twig into a river that (magically) always has exactly the same water flow patterns. If you were to drop a second twig just a fraction of an inch apart from the first, you might expect it to end up near the first one... and certainly closer to the first than if you had dropped it on the other side of the river. If the river were a linear system, this would be a good assumption. If it's chaotic,* it might still be a good assumption - but only for the first few dozen yards or so. If the river has a fork a mile downstream, predicting which branch the twig will take based only on its initial placement and the route the first one took is pretty much futile.
Returning from the river to an early 1960s lab:
Lorenz was trying to rerun a weather simulation. The numbers he entered as initial conditions for his model accidently happened to be very slightly different from the ones he used before... but they resulted in an entirely different weather pattern.
If Lorenz had stopped there, it might not have meant much. You didn't need a scientist to tell you weather was complex. Lorenz went further though. He spent quite a bit of time studying the mathematics and simplified his weather model to a set of three very simple equations that still resulted in very large sensitivity to initial conditions. What this meant was that at least part of the difficulty with long-term weather predictions was not due to the model being a poor explanation for reality, but to very small things (like a butterfly flapping its wings) having a surprisingly powerful effect on the system in the long-term.
This didn't really help long-term weather prediction all that much, as continuously monitoring every butterfly in the world unfortunately remains outside of the NOAA's budget limits. It has helped other fields where tighter control over the system was possible, though. Things that had previously seemed liked random noise in linear models, or problems that had been impossible to solve using linear models could be addressed with non-linear ones (and the help of massive computing power).
The key point is that, while chaotic systems sometimes seem random, they aren't. They are difficult to work with, but you can analyze them, both numerically (using a computer) and qualitatively (using plots usually made by a computer) to get useful information. As my differential equations text said after all the bad news, "We don't have to give up studying the solutions of the Lorenz equations. We just have to ask the right questions."
For instance, you might determine if using a much more convenient linear model under a particular region of a non-linear system is appropriate, and then use that simplified model to design something cool (Welcome to the world of engineering!). In a sense, chaos theory, despite the name, is really more about restoring order to previously unapproachable complexity.
*If you've ever dropped twigs in a river, it may come as no surprise that many natural systems are chaotic.
Moral: Run a find and replace, substituting "butterfly" for "seagull", before submitting academic papers for publication.
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RE: Edward Lorenz, Father of Chaos Theory, Dies - 
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