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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|0VGFHTM75Y]].\nThe map $f$ is *chaotic in the sense of Devaney* if:\n\n1. $f$ is [[topologically transitive|FR49Y9GQHR]].\n2. The set of [[periodic points|T75CZ5H47N]] of $f$ is [[dense|0BSJSX7ME2]] in $X$.\n3. $f$ has *sensitive dependence on initial conditions*: there is $\\delta \u003e 0$ such that each $x \\in X$ and $\\eta \u003e 0$ admit $y \\in X$ and $n \\ge 0$ as follows.\n   They satisfy $d(x, y) \u003c \\eta$ and $d(f^n(x), f^n(y)) \u003e \\delta$.\n\n## Remarks\n\nBanks, Brooks, Cairns, Davis and Stacey showed that if $X$ is infinite, then conditions 1 and 2 imply condition 3.\nExample: the doubling map $f(x) = 2x \\bmod 1$ on the circle $\\R / \\Z$.\nIt is transitive, since $f^n$ maps any arc of length greater than $2^{-n}$ onto the whole circle.\nIts periodic points are the points $k / (2^n - 1)$, which are dense.\nIt is sensitive with $\\delta = 1/8$: for $y \\neq x$ close to $x$, the distance between $f^n(x)$ and $f^n(y)$ doubles at each step while it is below $1/4$, so it eventually exceeds $1/8$.\nNon-example: the rotation $x \\mapsto x + \\alpha \\bmod 1$ with $\\alpha$ irrational is transitive but has no periodic points, so condition 2 fails; it is also not sensitive, being an isometry.\n\n## Explanations\n\n### L1 human\n\nA map is chaotic in the sense of Devaney if it links all regions, has repeating orbits near every point, and amplifies small differences in starting points.\n",
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