Suppose we note the s-values where successive period-doubling bifurcations occur.
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
|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.|
Return to Deterministic Chaos.