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  "body": "\n## Example\n\nFor a parameter $r \\in [0, 4]$, the *logistic map* is\n$$f_r \\colon [0, 1] \\to [0, 1], \\qquad f_r(x) = r x (1 - x).$$\nIt is a [[discrete-time dynamical system|5AA9AJEARJ]] on $[0, 1]$: the maximum of $f_r$ on $[0, 1]$ is $f_r(1/2) = r/4 \\le 1$ and $f_r \\ge 0$ there, so $f_r$ maps $[0, 1]$ into itself.\n\nCheck of the first bifurcations.\nThe [[fixed points|R6QVS6P21N]] are $0$ and, for $r \u003e 1$, $x_r = 1 - 1/r$.\nSince $f_r'(x) = r(1 - 2x)$, one gets $f_r'(0) = r$ and $f_r'(x_r) = 2 - r$.\nSo for $1 \u003c r \u003c 3$ the fixed point $x_r$ attracts nearby points.\nAt $r = 3$ its derivative is $-1$, and the map has a [[period-doubling bifurcation|NQDTTZ59X8]].\nFor $r$ slightly above $3$ an attracting orbit of period $2$ appears.\n\n## Remarks\n\nAs $r$ grows further, attracting orbits of periods $4, 8, 16, \\dots$ appear at parameters that accumulate at $r_\\infty \\approx 3.5699$.\nFor $r = 4$ the map is [[topologically conjugate|HNXXAPQKG3]] to the full [[tent map|D0A22MEB1F]] $T(y) = 1 - \\abs{1 - 2y}$ by $h(y) = \\sin^2(\\pi y / 2)$.\nIndeed $f_4(h(y)) = 4 \\sin^2(\\pi y/2) \\cos^2(\\pi y/2) = \\sin^2(\\pi y)$, and $h(T(y)) = \\sin^2(\\pi y)$ for $y \\le 1/2$ and $\\sin^2(\\pi - \\pi y) = \\sin^2(\\pi y)$ for $y \u003e 1/2$.\nFor $r = 4$ the measure with density $\\frac{1}{\\pi \\sqrt{x(1 - x)}}$ on $(0, 1)$ is invariant.\nThe map is the discrete-time analogue of the [[logistic equation|0CC4VC7KR3]] of population growth.\n\n## Explanations\n\n### L1 human\n\nThe logistic map is a toy model of a population from one year to the next: it grows when small and is held back when large.\nOne step takes just two multiplications and a subtraction.\nYet as $r$ increases the long-run behaviour goes from a steady state to a cycle of $2$ years, then $4$, $8$, and then to chaos.\nIt became famous as the simplest example of how chaos can arise from a very simple rule.\n",
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