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  "body": "\n## Summary\n\nA periodic solution of a difference equation is a solution that is a [[periodic sequence|Q3B6K9HYRG]].\nFor a first-order autonomous equation $x_{n+1} = f(x_n)$, it is the orbit of a [[periodic point|T75CZ5H47N]] of $f$.\nEquilibria are the periodic solutions of period $1$.\nPeriodic solutions, their periods and their stability describe much of the long-run behaviour of nonlinear recursions.\nTheir birth is studied through bifurcations such as period doubling.\nIt is the discrete counterpart of a [[periodic solution|FHNDX22WMK]] of a differential equation.\n\n## Explanations\n\n### L1 human\n\nA periodic solution is a sequence produced by the recursion that comes back to the same state after a fixed number of steps.\nFrom then on it repeats the same cycle forever.\nFor $x_{n+1} = 1 - x_n$ with $x_0 = 0$, the solution is $0, 1, 0, 1, \\dots$, a cycle of period $2$.\nIn the logistic map $x_{n+1} = r x_n (1 - x_n)$ with $r$ a little above $3$, solutions settle near a cycle of period $2$.\n",
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