# Deterministic Chaos

## 6.N. Period-Doubling Scaling and the Feigenbaum Constant

Suppose we note the s-values where successive period-doubling bifurcations occur.

Call s0 the value where the nonzero fixed point becomes unstable and a stable 2-cycle appears,
s1 the value where this 2-cycle becomes unstable and a stable 4-cycle appears,
s2 the value where this 4-cycle becomes unstable and a stable 8-cycle appears,
and so on.

The first few points are illustrated in the picture. Unfortunately, calculating the bifurcation values si to high accuracy is very difficult. Much easier is to determine the superstable values s*i, the s-values in each periodic window where x = 1/2 belongs to the cycle. This is a reasonable approach because the sequences of period-doubling and superstable values share many properties.

 Here are the first 14 supersable s*i

The simplest approach to detect a scaling relation between the s*n is to calculate how the dstance between them changes. That is, find the ratio of successive differences.

 Here are the ratios of the differnces of the first few supersable s*i.

It's interesting that the ratios of successive diatances converge, but is this enough to earn Feigenbaum a MacArthur grant? The real interest in this constant comes from its universality.

 There is another constant associated with this scaling.