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  "body": "\n## Problem\n\nConsider the [[Lorenz system|WYRC41CQN9]] in $\\R^3$,\n$$\\dot x = \\sigma (y - x), \\qquad \\dot y = x(\\rho - z) - y, \\qquad \\dot z = xy - \\beta z,$$\nwith the classical parameters $\\sigma = 10$, $\\rho = 28$ and $\\beta = 8/3$.\nDoes the flow of this system have a [[strange attractor|9WKCZGG906]], with the dynamics of the geometric Lorenz attractor?\n\n## Remarks\n\nThe geometric Lorenz attractor is a model flow, built in the 1970s from the features seen in numerical simulations.\nFor the model, the existence of a robust [[attractor|7FWX24X5MW]] with sensitive dependence on initial conditions was proved.\nThe question was whether the actual Lorenz equations behave like the model.\nSmale included it as the fourteenth problem of his 1998 list of problems for the next century.\nWarwick Tucker gave a positive answer with a computer-assisted proof, published in 2002.\nHis proof combines rigorous numerics with interval arithmetic, far from the origin, with normal form theory near the equilibrium at the origin.\n\n## Explanations\n\n### L1 human\n\nThe Lorenz system is a simple model of convection whose solutions trace the famous butterfly-shaped figure on a computer.\nPictures are not proofs: rounding errors might hide a stable cycle, and the figure might then be an illusion.\nSmale asked for a proof that the butterfly is a true chaotic attractor.\nThe answer is yes, and its proof was a major use of computers to settle a question in dynamics.\n",
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      "first_stated": "1998",
      "posed_by": "Stephen Smale"
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    "name": "Smale's fourteenth problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lorenz_system"
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        "url": "https://en.wikipedia.org/wiki/Smale%27s_problems"
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    "tag": "VDVQD2HFSD",
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  "msc": [
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  "name": "Smale's fourteenth problem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/VDV/VDVQD2HFSD.md",
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  "url": "https://tamnd.github.io/math2/wiki/smales-fourteenth-problem/",
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