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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\R^n$ and let $f \\colon \\R \\times U \\to \\R^n$ be continuous.\nA [[solution|07VX34QXAV]] $x \\colon \\R \\to U$ of $x' = f(t, x)$ is periodic if there is a real number $T \u003e 0$ with\n$$x(t + T) = x(t) \\quad \\text{for all } t \\in \\R.$$\nSuch a $T$ is a period of $x$.\nIf $f$ is $T$-periodic in $t$, that is, $f(t + T, x) = f(t, x)$ for all $t$ and $x$, a $T$-periodic solution is a periodic solution with period $T$.\n\n## Remarks\n\nA periodic solution is a [[periodic function|0ATF6Q0633]] of $t$ that solves the equation.\nFor an autonomous equation $x' = f(x)$, each constant solution is periodic with every period, and the trajectory of a nonconstant periodic solution is a [[periodic orbit|0AA97XA70K]].\nExample: for $x' = -x + \\cos t$, the function $x(t) = \\frac{1}{2}(\\cos t + \\sin t)$ has $x'(t) = \\frac{1}{2}(\\cos t - \\sin t) = -x(t) + \\cos t$.\nIt is a $2\\pi$-periodic solution of an equation with $2\\pi$-periodic right-hand side.\nNon-example: the other solutions $x(t) = \\frac{1}{2}(\\cos t + \\sin t) + c e^{-t}$ with $c \\ne 0$ are not periodic, because $c e^{-t}$ is not.\nA maximal solution of a $T$-periodic equation with $x(t_0 + T) = x(t_0)$ for one $t_0$ is $T$-periodic when solutions are unique.\n\n## Explanations\n\n### L1 human\n\nA periodic solution returns to the same state after a fixed time and repeats forever.\nFor a periodically forced system, the most interesting periodic solutions have the same period as the forcing.\n",
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