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What Exactly is Chaos?
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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Update: June 2012
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