```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 theform f(x,y) = (y,g(x,y)) for different functions g and differentinitial values of x and y.For example let g(x,y) = 2^y - xf 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 goesround the point (1,1) in a loop if a is close to 0.  As we increasethe size of a the loop seems to get 'pulled' towards the other fixedpoint (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 extremelyquickly.  This seems to be chaotic: Although the same behaviour occursif we increase a further, the number of times it goes around the loopbefore it shoots off is unpredictable.Does anybody know anything about these functions?Is there a general theory of them?
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