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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval and let $f_\\mu \\colon X \\to X$, $\\mu \\in I$, be a family of maps of a set $X \\subseteq \\R$ with $(\\mu, x) \\mapsto f_\\mu(x)$ smooth.\nA *period-doubling cascade* of the family is a strictly increasing sequence of parameters $\\mu_1 \u003c \\mu_2 \u003c \\mu_3 \u003c \\cdots$ in $I$ with a limit $\\mu_\\infty \\in I$, such that for each $k \\ge 1$:\n\n1. for $\\mu$ slightly less than $\\mu_k$, the family has an attracting [[periodic solution|3W358BSRN2]] of prime period $2^{k-1}$;\n2. at $\\mu = \\mu_k$ this cycle undergoes a [[period-doubling bifurcation|NQDTTZ59X8]]: it loses stability, and an attracting cycle of prime period $2^k$ is born from it.\n\n## Remarks\n\nExample: the logistic family $f_r(x) = r x (1 - x)$ on $[0, 1]$ has such a cascade with $r_1 = 3$ and $r_2 = 1 + \\sqrt{6} \\approx 3.449$.\nAt $r_1 = 3$ the fixed point $1 - 1/r$ has multiplier $f_r'(1 - 1/r) = 2 - r = -1$, and a $2$-cycle appears.\nThe limit is $r_\\infty \\approx 3.5699$.\n\nNon-example: the family $f_\\mu(x) = \\mu x$ on $\\R$ has no period-doubling cascade.\nIts only possible attracting cycle is the fixed point $0$ for $\\abs{\\mu} \u003c 1$, and it never has an attracting cycle of period $2$, so clause 2 fails already for $k = 1$.\n\nFor typical smooth unimodal families with a quadratic maximum, such as the logistic family, the ratios $(\\mu_k - \\mu_{k-1})/(\\mu_{k+1} - \\mu_k)$ tend to the Feigenbaum constant $\\delta \\approx 4.6692$.\n",
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