This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.

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.

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.

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