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 a1 and a2 for superstable
2-cycle and superstable 4-cycle.
The next picture shows the relative placements of a1, a2,
a3, and a4 in the bifurcation diagram, with the horizontal
scale suitably expanded to make these distances visible.
More numerical experiments indicate the sequence of
ratios an/an+1 approaches a limit, the
Feigenbaum alpha constant 2.502907876... .
Return to the Feigenbaum Constant.
