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  "body": "\n## Summary\n\nThe $\\omega$-limit set of a point is the set of states that its solution approaches again and again as time goes to $+\\infty$.\nThe $\\alpha$-limit set is the same notion for time going to $-\\infty$.\nLimit sets describe the long-time behaviour of a dynamical system: they can be equilibria, [[periodic orbits|0AA97XA70K]] or more complicated sets.\nThey are closed [[invariant sets|WWZTW11664]], and for a bounded forward orbit the $\\omega$-limit set is nonempty, compact and connected.\n\n## Explanations\n\n### L1 human\n\nRun the system for a long time and watch where the state keeps returning.\nThose places form the $\\omega$-limit set.\nFor the equation $x' = -x$, every solution tends to $0$, so the $\\omega$-limit set of every point is $\\set{0}$.\nFor a system whose solutions spiral onto the unit circle, the $\\omega$-limit set of every point except the origin is the whole circle.\nIt is the \"final destination\" of the motion, even when the motion never stops.\n",
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