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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]] $x' = f(x)$.\nLet $x \\in U$ be such that the solution through $x$ is defined for all $t \\ge 0$.\nThe $\\omega$-limit set of $x$ is the set $\\omega(x)$ of points $y \\in U$ for which there is a sequence $t_k \\to +\\infty$ with $\\varphi_{t_k}(x) \\to y$.\nLet the solution through $x$ be defined for all $t \\le 0$.\nThe $\\alpha$-limit set $\\alpha(x)$ is the set of points $y \\in U$ with $\\varphi_{t_k}(x) \\to y$ for some sequence $t_k \\to -\\infty$.\n\n## Remarks\n\nEquivalently, $\\omega(x) = \\bigcap_{T \\ge 0} \\overline{\\set{\\varphi_t(x) : t \\ge T}} \\cap U$, where the bar denotes the closure in $\\R^n$.\nThe set $\\omega(x)$ is closed in $U$ and [[invariant|WWZTW11664]].\nIf the forward orbit of $x$ lies in a compact subset of $U$, then $\\omega(x)$ is nonempty, compact and connected, and $\\varphi_t(x)$ tends to $\\omega(x)$ as $t \\to \\infty$.\nExample: in polar coordinates, let $r' = r(1 - r^2)$ and $\\theta' = 1$ in the plane.\nFor $x \\ne 0$ the radius tends to $1$ while the angle keeps turning.\nSo every point of the unit circle is a limit of $\\varphi_{t_k}(x)$ for suitable $t_k \\to \\infty$, and $\\omega(x)$ is the unit circle.\nFor $0 \u003c \\abs{x} \u003c 1$, the radius tends to $0$ as $t \\to -\\infty$, so $\\alpha(x) = \\set{0}$.\nNon-example: for $x' = 1$ on $\\R$, the solution $\\varphi_t(x) = x + t$ has no convergent sequence $\\varphi_{t_k}(x)$ with $t_k \\to \\infty$, so $\\omega(x)$ is empty.\nThis is possible because the forward orbit is not bounded.\n\n## Explanations\n\n### L1 human\n\nThe $\\omega$-limit set collects all points that the forward motion comes close to again and again.\nThe $\\alpha$-limit set does the same for the motion run backwards in time.\n",
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