# Deterministic Chaos

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

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

Call s_{0} the value where the nonzero fixed
point becomes unstable and a stable 2-cycle appears,

s_{1} the value
where this 2-cycle becomes unstable and a stable 4-cycle appears,

s_{2} the value where this 4-cycle becomes unstable and a stable
8-cycle appears,

The first few points are illustrated in the picture.

Unfortunately, calculating the bifurcation values s_{i}
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.

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.

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