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  "body": "\n## Description\n\nThe Feigenbaum constants are two real numbers that describe the cascade of [[period-doubling bifurcations|NQDTTZ59X8]] of a one-parameter family of unimodal maps with a quadratic maximum.\nLet $f_r$ be such a family, for example the [[logistic map|QN9X9BNKTX]] $f_r(x) = r x (1 - x)$.\nLet $r_n$ be the parameter at which the attracting orbit of period $2^{n-1}$ loses stability and an attracting orbit of period $2^n$ appears.\nFor the logistic map $r_1 = 3$ and $r_2 = 1 + \\sqrt 6 \\approx 3.4495$.\nThe first Feigenbaum constant is\n$$\\delta = \\lim_{n \\to \\infty} \\frac{r_n - r_{n-1}}{r_{n+1} - r_n} = 4.669201609\\ldots.$$\nLet $c$ 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 $2^n$.\nLet $d_n = f_{R_n}^{2^{n-1}}(c) - c$, the signed distance from $c$ to the point of its orbit halfway around the cycle.\nThe second Feigenbaum constant is\n$$\\alpha = \\lim_{n \\to \\infty} \\abs{\\frac{d_n}{d_{n+1}}} = 2.502907875\\ldots.$$\n\n## Remarks\n\nCheck for the logistic map: the period-$2$ orbit is stable while $\\abs{(f_r^2)'} \u003c 1$ on it, and solving gives $r_2 = 1 + \\sqrt 6$; the parameters $r_n$ accumulate at $r_\\infty \\approx 3.5699$.\nThe ratios $d_n / d_{n+1}$ alternate in sign, which is why $\\alpha$ is often given as $-2.5029\\ldots$.\nThe constants are universal: they take the same values for every smooth family of unimodal maps with a nondegenerate quadratic maximum that has such a cascade.\nFeigenbaum explained this by a renormalization operator acting on maps of the interval, with a fixed point whose unstable eigenvalue is $\\delta$ and whose scaling factor is $\\alpha$.\nLanford gave a computer-assisted proof of the existence of this fixed point in 1982.\nIt is not known whether $\\delta$ or $\\alpha$ is irrational.\n\n## Explanations\n\n### L1 human\n\nWhen a parameter is turned up, many systems double their period again and again: a cycle of $2$, then $4$, $8$, and so on, until chaos.\nThe gaps between these doublings shrink, each about $4.669$ times smaller than the one before.\nThe same number appears for dripping taps, electronic circuits and many simple formulas, which made it a surprise when Feigenbaum found it in the 1970s.\n",
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