Feigenbaum function

In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:
Feigenbaum-Cvitanović functional equation.
This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by Mitchell Feigenbaum and Predrag Cvitanović, the equation is the mathematical expression of the universality of period doubling. It specifies a function "g" and a parameter by the relation
with the initial conditions
For a particular form of solution with a quadratic dependence of the solution
near is one of the Feigenbaum constants.
Scaling function.
The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade. The attractor is a Cantor set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size "dn". For a fixed "dn" the set of segments forms a cover "Δn" of the attractor. The ratio of segments from two consecutive covers, "Δn" and "Δn+1" can be arranged to approximate a function "σ", the Feigenbaum scaling function.