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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[continuous map|0X3WGX95FE]], with [[iterates|AYVTR2ES5B]] $f^n$.\nThe map $f$ has *sensitive dependence on initial conditions*, the butterfly effect, if there is a constant $\\delta \u003e 0$ with the following property.\nFor every $x \\in X$ and every $\\varepsilon \u003e 0$, there are a point $y \\in X$ with $d(x, y) \u003c \\varepsilon$ and an integer $n \\ge 0$ with $d(f^n(x), f^n(y)) \u003e \\delta$.\n\n## Remarks\n\nThe constant $\\delta$ is the same for all $x$; the point $y$ and the time $n$ depend on $x$ and $\\varepsilon$.\nFor a flow $(\\varphi_t)$ one asks instead for a time $t \\ge 0$ with $d(\\varphi_t(x), \\varphi_t(y)) \u003e \\delta$.\nSensitive dependence is one of the conditions in Devaney's definition of chaos.\n\nExample: the doubling map $f(x) = 2x \\bmod 1$ on the circle $\\R / \\Z$, with the distance $d$ to the nearest integer of the difference.\nTake $\\delta = 1/8$.\nGiven $x$ and $\\varepsilon \u003e 0$, choose $n$ with $2^{-n} \u003c \\varepsilon$ and let $y = x + 2^{-n-2}$.\nThen $d(x, y) \u003c \\varepsilon$, and $f^n(y) - f^n(x) = 1/4 \\bmod 1$, so $d(f^n(x), f^n(y)) = 1/4 \u003e \\delta$.\n\nNon-example: the rotation $f(x) = x + \\alpha \\bmod 1$ on $\\R / \\Z$ is an isometry, so $d(f^n(x), f^n(y)) = d(x, y)$ for all $n$.\nPoints closer than $\\delta$ stay closer than $\\delta$ forever, so no $\\delta \u003e 0$ works.\n",
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    "name": "Butterfly effect",
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