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## Summary

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

$g(x)=-\alpha g(g(-x/\alpha ))$

with the initial conditions

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

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.

The power series of $g$  is approximately

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

## Renormalization

The Feigenbaum function can be derived by a renormalization argument.

The Feigenbaum function satisfies

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

for any map on the real line $F$  at the onset of chaos.

## 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.