# Deterministic Chaos

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

## The Feigenbaum Alpha Constant

There is another quantitative feature common to these period-doubling
bifurcation sequences:

the limit of the ratios of the distance from
x = 1/2 (the critical point of the logistic map)
and the point nearest it in the superstable cycle.

Here we see the distances a_{1} and a_{2} for superstable
2-cycle and superstable 4-cycle.
The next picture shows the relative placements of a_{1}, a_{2},
a_{3}, and a_{4} in the bifurcation diagram, with the horizontal
scale suitably expanded to make these distances visible.

More numerical experiments indicate the sequence of
ratios a_{n}/a_{n+1} approaches a limit, the
Feigenbaum alpha constant 2.502907876... .

Return to the Feigenbaum Constant.