Pictorial Renormalization Group - Demo 8


For a=3.45

the slope of f 2 at the fixed point in the rescaled, truncated plot of f 2 has become less than -1 and the fixed point is unstable. A small 2-cycle develops in f 2. (Run long enough to eliminate transients and hit "Clear" to erase old iterations, or set "trans" to 100.)

This can be verified looking at f 4(x)=f 2(f 2(x)) i.e. the functional composition of f 2

where we see that the slope of f 4 is indeed greater than unity at the fixed point of f 2, and two new fixed points develop in f 4, corresponding to the two points of the 2-cycle in f 2.

In the original plot of the map f:

this corresponds to a 4-cycle, i.e. 4 different values of x are visited after transients have died out.

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Last modified Sunday, December 12, 1999
Michael Cross