# Pictorial Renormalization Group - Demo 8

## a=3.45

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