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.