Feigenbaum constants

Description

The Feigenbaum constants are two real numbers that describe the cascade of period-doubling bifurcations of a one-parameter family of unimodal maps with a quadratic maximum. Let $f_r$ be such a family, for example the logistic map $f_r(x) = r x (1 - x)$. Let $r_n$ be the parameter at which the attracting orbit of period $2^{n-1}$ loses stability and an attracting orbit of period $2^n$ appears. For the logistic map $r_1 = 3$ and $r_2 = 1 + \sqrt 6 \approx 3.4495$. The first Feigenbaum constant is $$\delta = \lim_{n \to \infty} \frac{r_n - r_{n-1}}{r_{n+1} - r_n} = 4.669201609\ldots.$$ Let $c$ be the critical point, and let $R_n$ be the parameter, between $r_n$ and $r_{n+1}$, at which $c$ is periodic of least period $2^n$. Let $d_n = f_{R_n}^{2^{n-1}}(c) - c$, the signed distance from $c$ to the point of its orbit halfway around the cycle. The second Feigenbaum constant is $$\alpha = \lim_{n \to \infty} \abs{\frac{d_n}{d_{n+1}}} = 2.502907875\ldots.$$