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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 Lyapunov stable if for each $\\varepsilon \u003e 0$ there is a $\\delta \u003e 0$ with this property. For each $x_0 \\in U$ with $\\|x_0 - x^*\\| \u003c \\delta$, the solution $\\varphi_t(x_0)$ exists for all $t \\ge 0$, and $\\|\\varphi_t(x_0) - x^*\\| \u003c \\varepsilon$ for all $t \\ge 0$.\nThe point $x^*$ is unstable if it is not Lyapunov stable.\n\n## Remarks\n\nFor $x' = -x$ and for $x' = 0$ on $\\mathbb{R}$, the equilibrium $0$ is stable. For $x' = x$, it is unstable.\nThe centre of the [[harmonic oscillator|0YG0ZJPTWG]] is stable but not [[asymptotically stable|05Y1JAM3Y5]]: nearby solutions stay near, but they do not approach it.\nA [[Lyapunov function|04P7B5XC22]] gives a way to prove stability without solving the equation.\n\n## Explanations\n\n### L1 human\n\nAn equilibrium is stable if solutions that start close to it stay close to it forever.\nTo stay within distance $\\varepsilon$, it is enough to start within a smaller distance $\\delta$.\n",
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