# Pictorial Renormalization Group - Demo 10

## a=3.498 - 3.5546

The sequence of behavior should now be apparent. As we have seen at *a=3.498* we have a super stable 4-cycle, which corresponds to a superstable fixed point in *f*^{ 4}. We need only study the behavior in the region near *x=½* given by scaling and truncating by putting *nsc=2*.

Increasing to *a=3.545* the height of the curve goes up, the slope at the fixed point decrease below -1 and the fixed point goes unstable to a 2-cycle in *f*^{ 4}, i.e. an 8-cycle in *f*.

At *a=3.5546* this becomes a super stable cycle:

shown as superstable fixed points in *f*^{ 8}=f^{ 4}(f^{ 4}(x))

We can focus on the central region be enlarging and inverting (increasing *nsc* to 3):

and we are back to the same situation as in the first applet except:
- we have increased
*a*;
- we have performed an additional functional composition
*f*^{ 4} -> f^{ 8}=f^{ 4}(f^{ 4}(x)) so that the fixed point corresponds to an extra period doubling of the original map iteration;
- we have rescaled both axes by a factor of -2.502907875... and truncated to the range
*|x-½|=0.5*.

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Last modified Sunday, December 12, 1999

Michael Cross