This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet 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 computations by F. Christiansen, P. Cvitanovic and H.H. Rugh,

it has the value 4.6692016091029906718532038... "Note": several books

have published incorrect values starting 4.6692016_6"...; 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

2.50290787509589282228390287272909. 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[a] is the algebraic distance between nearest elements

of the attractor cycle of period 2^a, then d[a]/d[a+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. A" 24 (1991), pp. 3363-3368.

4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the

period-doubling operator in terms of cycles", "J. Phys A" 23, L713

(1990).

5. M. Feigenbaum, The Universal Metric Properties of Nonlinear

Transformations, "J. Stat. Phys" 21 (1979), p. 69.

6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, "Los

Alamos Sci" 1 (1980), pp. 1-4. Reprinted in "Universality in

Chaos", compiled by P. Cvitanovic.

Feigenbaum Constants

http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html

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