Ask A Scientist Mathematics Archive

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.
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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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