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 4.66920166...; 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 2.50290787509589282228390287272909. 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