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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be locally Lipschitz. Let $\\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \\in U$ be an [[equilibrium point|0G4X36AE0X]].\nThe point $x^*$ is asymptotically stable if it is [[Lyapunov stable|092S6VBECQ]] and there is an $\\eta \u003e 0$ with this property. For each $x_0 \\in U$ with $\\|x_0 - x^*\\| \u003c \\eta$, $\\varphi_t(x_0) \\to x^*$ as $t \\to \\infty$.\nIf $U = \\mathbb{R}^n$ and $\\varphi_t(x_0) \\to x^*$ for each $x_0 \\in \\mathbb{R}^n$, and $x^*$ is Lyapunov stable, then $x^*$ is globally asymptotically stable.\n\n## Remarks\n\nFor $x' = -x$ on $\\mathbb{R}$, the equilibrium $0$ is globally asymptotically stable, since $\\varphi_t(x_0) = x_0 e^{-t}$.\nThe point $x^*$ is exponentially stable if there are $\\eta, C, \\alpha \u003e 0$ with $\\|\\varphi_t(x_0) - x^*\\| \\le C e^{-\\alpha t} \\|x_0 - x^*\\|$ for $\\|x_0 - x^*\\| \u003c \\eta$ and $t \\ge 0$.\nThe attraction condition alone does not imply Lyapunov stability. There are planar examples where all solutions tend to $x^*$, but some first make a large excursion.\n\n## Explanations\n\n### L1 human\n\nAn equilibrium is asymptotically stable if nearby solutions stay near it and also approach it in the long run.\nA ball at the bottom of a bowl with friction is a typical example.\n",
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