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  "body": "\n## Summary\n\nThe butterfly effect is the popular name for sensitive dependence on initial conditions in a dynamical system.\nArbitrarily small changes in the initial state can lead to large differences in the later states.\nThe name comes from Edward Lorenz, whose computer model of convection, now the [[Lorenz system|WYRC41CQN9]], showed this behaviour.\nIt is a central ingredient of [[chaos theory|DBKXCGDKTD]] and sets practical limits on long-range prediction, even for fully deterministic systems.\n\n## Explanations\n\n### L1 human\n\nThe image is that the flap of a butterfly's wings could, weeks later, change whether a storm forms far away.\nThe point is not that butterflies cause storms, but that tiny causes can have effects that grow out of all proportion.\nA simple model is the doubling map $x \\mapsto 2x \\bmod 1$ on $[0, 1)$.\nTwo starting values that differ by $10^{-6}$ differ by about $2^n \\cdot 10^{-6}$ after $n$ steps, as long as this is small.\nAfter about $20$ steps the difference is of the same size as the whole interval.\n",
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