Timsn274
Posts:
243
Registered:
12/13/04
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Re: Non-linear recursive functions
Posted:
Jul 8, 2012 7:07 PM
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On Wednesday, July 4, 2012 3:42:56 PM UTC-4, Richard Clark wrote:
> On Wednesday, July 4, 2012 2:05:46 AM UTC+1, Tim Norfolk wrote:
> > 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'.
>
> Thanks, but I don't think ergodic theory applies to the function above because Lesbesgue measure isn't invariant under it (although I think it may be quasi-invariant).
If this were my problem, I might start with a local linearization of the operator, to determine which of the fixed points were attractive.
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