Feigenbaum's constant

Q10: What is Feigenbaum's constant?
A10: In a period doubling cascade, such as the logistic equation, consider the
parameter values where period-doubling events occur (e.g. r[1]=3, r[2]=3.45,
r[3]=3.54, r[4]=3.564...).  Look at the ratio of distances between consecutive
doubling parameter values; let delta[n] = (r[n+1]-r[n])/(r[n+2]-r[n+1]).  Then
the limit as n goes to infinity is Feigenbaum's (delta) constant.

Based on independent computations by Jay Hill and Keith Briggs, it has the
value 4.669201609102990671853...  Note: several books have published incorrect
values starting 4.66920166...; the last repeated 6 is a typographical error.

The interpretation of the delta constant is as you approach chaos, each
periodic region is smaller than the previous by a factor approaching 4.669...
Feigenbaum's constant is important because it is the same for any function or
system that follows the period-doubling route to chaos and has a one-hump
quadratic maximum.  For cubic, quartic, etc. there are different Feigenbaum
constants.

Feigenbaum's alpha constant is not as well known; it has the value
2.502907875095.  This constant is the scaling factor between x values at
bifurcations.  Feigenbaum says, "Asymptotically, the separation of adjacent
elements of period-doubled attractors is reduced by a constant value [alpha]
from one doubling to the next".  If d[n] is the algebraic distance between
nearest elements of the attractor cycle of period 2^n, then d[n]/d[n+1]
converges to -alpha.

References:

1.  K. Briggs, How to calculate the Feigenbaum constants on your PC, _Aust.
Math.  Soc.  Gazette_ 16 (1989), p. 89.

2.  K. Briggs, A precise calculation of the Feigenbaum constants, _Mathematics
of Computation_ 57 (1991), pp. 435-439.

3.  K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for Mandelsets,
_J. Phys._ A24 (1991), pp. 3363-3368.

4.  M. Feigenbaum, The Universal Metric Properties of Nonlinear
Transformations, _J. Stat. Phys_ 21 (1979), p. 69.

5.  M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los Alamos Sci_
1 (1980), pp. 1-4.  Reprinted in _Universality in Chaos_ , compiled by P.
Cvitanovic.