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?