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=3, r=3.45, r=3.54, r=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.