Date: Jul 3, 2012 5:30 PM
Author: Richard Clark
Subject: Non-linear recursive functions
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?