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28 What is Feigenbaum's constant?




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This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.

28 What is Feigenbaum's constant?

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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