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It models animal populations. The equation is x -> c*x*(1-x), where x is the population (between 0 and 1) and c is a growth constant. Iteration of this equation yields the period doubling route to chaos. For c between 1 and 3, the population will settle to a fixed value. At 3, the period doubles to 2; one year the population is very high, causing a low population the next year, causing a high population the following year. At 3.45, the period doubles again to 4, meaning the population has a four year cycle. The period keeps doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At 3.57, chaos occurs; the population never settles to a fixed period. For most c values between 3.57 and 4, the population is chaotic, but there are also periodic regions. For any fixed period, there is some c value that will yield that period. See "An Introduction to Chaotic Dynamical Systems" for more information.
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.