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  "body": "\n## Summary\n\nThe Poincaré map, or first return map, belongs to a [[periodic orbit|0AA97XA70K]] and a cross-section of it.\nIt follows nearby solutions once around the orbit until they hit the cross-section again.\nIt replaces a continuous-time flow in dimension $n$ by a discrete map in dimension $n - 1$.\nThe periodic orbit becomes a fixed point of the map, and the stability of the orbit is the stability of this fixed point.\nThe idea is due to Henri Poincaré and is a basic tool in the study of periodic orbits, bifurcations and chaos.\n\n## Explanations\n\n### L1 human\n\nPut a small screen across a closed loop traced by the system.\nStart a solution on the screen and wait until it crosses the screen again; the new crossing point is the image of the old one.\nIf the screen points come closer and closer to the loop, the loop attracts nearby motions.\nFor motions that spiral onto the unit circle, each crossing of the positive $x$-axis lies closer to the point $(1, 0)$ than the one before.\n",
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