## Subject: 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 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:

- K. Briggs, How to calculate the Feigenbaum constants on your PC,
*Aust.
Math. Soc. Gazette* 16 (1989), p. 89.
- K. Briggs, A precise calculation of the Feigenbaum constants,
*Mathematics
of Computation* 57 (1991), pp. 435-439.
- K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for Mandelsets,
*J. Phys. A* 24 (1991), pp. 3363-3368.
- 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).
- M. Feigenbaum, The Universal Metric Properties of Nonlinear Transformations,
*J. Stat. Phys* 21 (1979), p. 69.
- 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