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):

1. we have increased a;
2. 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;
3. we have rescaled both axes by a factor of -2.502907875... and truncated to the range |x-½|=0.5.