In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:^[1]

• the solution to the Feigenbaum-Cvitanović functional equation; and
• the scaling function that described the covers of the attractor of the logistic map

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,^[2] the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter α by the relation

${\displaystyle g(x)=-\alpha g(g(-x/\alpha ))}$

with the initial conditions

$${\displaystyle {\begin{cases}g(0)=1,\\g'(0)=0,\\g''(0)<0.\end{cases))}$$

The power series of ${\displaystyle g}$ is approximately^[3]

$${\displaystyle g(x)=1-1.52763x^{2}+0.104815x^{4}+0.026705x^{6}+O(x^{8})}$$

Renormalization

The Feigenbaum function can be derived by a renormalization argument.^[4]

The Feigenbaum function satisfies^[5]

$${\displaystyle g(x)=\lim _{n\rightarrow \infty }{\frac {1}{F^{\left(2^{n}\right)}(0)))F^{\left(2^{n}\right)}\left(xF^{\left(2^{n}\right)}(0)\right)}$$

${\displaystyle F}$

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 d_n. For a fixed d_n the set of segments forms a cover Δ_n of the attractor. The ratio of segments from two consecutive covers, Δ_n and Δ_n+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.