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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\R^n$, let $f \\colon U \\to \\R^n$ be locally Lipschitz, and let $\\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.\nFor $x \\in U$ let $I(x)$ be the maximal interval of existence of the solution through $x$.\nA set $M \\subseteq U$ is invariant if $\\varphi_t(x) \\in M$ for every $x \\in M$ and every $t \\in I(x)$.\nIt is positively invariant if $\\varphi_t(x) \\in M$ for every $x \\in M$ and every $t \\in I(x)$ with $t \\ge 0$, and negatively invariant if the same holds for $t \\le 0$.\n\n## Remarks\n\nA set is invariant exactly when it is a union of [[trajectories|07TZ2Q4C1M]].\nExample: for $x' = -y$, $y' = x$ on $\\R^2$, the flow is rotation by the angle $t$, so it preserves $x^2 + y^2$; hence each disc $\\set{x^2 + y^2 \\le r^2}$ is invariant.\nNon-example: for $x' = -x$ on $\\R$, the interval $[0, 1]$ is positively invariant, since $\\varphi_t(x) = e^{-t} x$ stays in $[0, 1]$ for $t \\ge 0$.\nIt is not invariant, since $\\varphi_{-1}(1) = e \u003e 1$.\nAn equilibrium point and a periodic orbit are invariant sets.\nFor a continuously differentiable function $V \\colon U \\to \\R$ with $\\nabla V \\cdot f \\le 0$ on $U$, each sublevel set $\\set{x \\in U : V(x) \\le c}$ is positively invariant.\nFor a general flow $\\varphi$ on a set $X$ the same definition reads $\\varphi_t(M) \\subseteq M$ for all $t$.\n\n## Explanations\n\n### L1 human\n\nAn invariant set is a trap with no way out in either direction of time.\nA positively invariant set is a trap that solutions can enter but never leave.\n",
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