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Name: Unknown
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993


Question:
What is the actual equation for Chaos? I have read many letters about this topic, and all simply talk about the how strange Chaos is. Can someone give me a definition of it, and maybe tell me what people are using it for.



Replies:
I think there are several definitions, so I will just mention the one I am familiar with. The basic idea is that if you have a deterministic function describing the evolution of a system x'=f(x) (x being some multi- dimensional variable) then you would expect you could determine the future behavior to any degree of accuracy, given that you know x accurately enough. (The original example was a variable x describing the weather at various locations - sounds like a good chaotic system, huh?) Well, it turns out that you can do this in a practical way for many functions F. But for an awful lot of them, this just is not possible. If you had picked a point x_2 instead of x, differing from x by some distance d, then the distance between the results of successive iterations (f(x_2) - f(x), f(f(x_2)) - f(f(x)), etc.) grows geometrically in a chaotic system, while it does not grow or can even shrink in a non-chaotic system. The geometric, or exponential growth of this "error" means that prediction is practically impossible beyond a certain number of iterations into the future. So, they cannot predict the weather beyond about a week ahead, simply because the accuracy in initial data required is impossibly high. Another aspect of chaos is that there is in fact quite a bit of "order" within it. In a nonchaotic system, the images of x under f follow an orbit (or are attracted to an orbit) which consists of a simple periodic figure (like the circular orbits of the planets). The most general multidimensional form is a "torus", where several different periods may compete in a quasiperiodic fashion on the same orbit. In a chaotic system, the points also can be following an orbit, but it has no periodicity in it. The simplest 1-dimensional example is f(x) = lambda - x^2, which, as you lower lambda from 1, goes through a series of transitions in which the stable orbits go from period 1 to period 2 to period 4 (successively doubling) until at a particular lambda the period goes to infinity, and the stable orbit is chaotic.

Arthur Smith



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