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Topic: Non-linear recursive functions
Replies: 6   Last Post: Jan 26, 2013 2:24 PM

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Posts: 263
Registered: 12/13/04
Re: Non-linear recursive functions
Posted: Jul 3, 2012 9:05 PM
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On Tuesday, July 3, 2012 5:30:09 PM UTC-4, Richard Clark wrote:
> I've been investigating orbits produced by iterating funtions of the
> form f(x,y) = (y,g(x,y)) for different functions g and different
> initial values of x and y.
> For example let g(x,y) = 2^y - x
> f then has 2 fixed points; at (1,1) and (2,2)
> (This is quite easy to do in Excel.)
> If we start from the point (1+a,1+a) where 0 < a < 1 the orbit goes
> round the point (1,1) in a loop if a is close to 0. As we increase
> the size of a the loop seems to get 'pulled' towards the other fixed
> point (2,2) so that it has a pear shape. As a gets very close to 1
> (e.g. 0.999) an interesting thing happens: The orbit goes round (1,1)
> in a loop a certain number of times and then shoots off extremely
> quickly. This seems to be chaotic: Although the same behaviour occurs
> if we increase a further, the number of times it goes around the loop
> before it shoots off is unpredictable.
> Does anybody know anything about these functions?
> Is there a general theory of them?

Look up 'ergodic theory'.

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