Pictorial Renormalization Group - Demo 11

Superstable cycles

Now, having displayed the intermediate structure, we can simply follow the scaling behavior of the superstable cycles i.e. we

• increase a to the value of the next 2 ⁿ superstable cycle
• increase the order of the functional composition to f ^{2 n}
• increase the order of the scaling to ⁿ

It should be apparent that to high accuracy the shape of the plotted curve does not change under this process. It is this invariance that leads to the scaling behavior.

At the final level of the table n=6 we have a 64-cycle. This is shown by undoing the scaling to show the full unit interval for f ⁶⁴:

which must show 64 intersection that are the stable fixed points (as well as intersections corresponding to the unstable fixed points); or returning to the original map and iterating

we see a 64-cycle.

This process of increasing n can of course be continued indefinitely, so that approaching the accumulation point a_c=3.5699456... orbits of arbitrarily large period can be found.