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  "body": "\n## Summary\n\nAn invariant set of a differential equation is a set of states that no solution can leave or enter.\nA solution that starts in the set stays in it for all times.\nEquilibria, [[periodic orbits|0AA97XA70K]], limit sets and stable manifolds are invariant sets.\nA positively invariant set only needs to be kept for positive times.\nSuch sets are used to trap solutions and to prove stability and existence of periodic orbits.\n\n## Explanations\n\n### L1 human\n\nPicture the state of a system moving along its [[trajectory|07TZ2Q4C1M]] in the phase plane.\nA set is invariant if every trajectory that touches it lies entirely inside it.\nFor the rotation $x' = -y$, $y' = x$, every disc $x^2 + y^2 \\le r^2$ is invariant, since each trajectory is a circle around the origin.\nThe half-plane $x \\ge 0$ is not invariant, because the circles leave it.\nA union of whole trajectories is always invariant.\n",
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