# 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:

• 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 , the equation is the mathematical expression of the universality of period doubling, it specifies a function g and a parameter α by the relation

$g(x)=-\alpha g(g({\frac {1}{\alpha }}x))$ with the initial conditions

• g(0) = 1,
• g′(0) = 0, and
• g′′(0) < 0

For a particular form of solution with a quadratic dependence of the solution near x=0, α=2.5029... 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.