Is this chaos?

[bob000555]

Let's say we define a set [math]S[/math] in the following manner.

 

[math] 0 < S_0 < 1 [/math]

[math] S_n = (1-S_{n-1})^x[/math]

 

 

If we take the example [math]S_0 = \frac{1}{2}[/math] and [math]x = 3[/math] then the set apears to not converge and to follow no set pateren, is this chaos?

Edited May 3, 2009 by bob000555

[Bignose]

I doesn't appear to converge in that it is tending toward a single number, but it is a converging oscillation between 0 and 1.

 

This isn't chaos in a proper mathematical term. Chaos mathematically is when a small change in inputs makes dramatic changes in outputs.

 

E.g. consider some function f(x)

 

If f(1.0)= 10 most functions behave reasonably well in that

f(1.0001) = something like 10.04 or 9.95 or even just 11 or 9. and f(1.0002) would then be 10.1 or 9.9 etc.

 

A chaotic function would be something like f(1.0) = 10, f(1.0001) = -765.9, f(1.0002) = 387.9, f(1.0003) = 22.8, f(1.0004) = -98.11, etc. Where a very small change in input leads to a very large change in output.

 

This is almost exclusively due to nonlinearity of the function (I don't know of a single linear chaotic example).

 

This is a decent article: http://www.imho.com/grae/chaos/chaos.html