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  "body": "\n## Summary\n\nA periodic solution of a differential equation is a solution defined for all times that repeats itself after a fixed time $T \u003e 0$.\nFor an autonomous system, a periodic solution that is not constant traces a [[periodic orbit|0AA97XA70K]] in phase space.\nFor an equation whose right-hand side is $T$-periodic in time, such as a periodically forced oscillator, one looks for solutions with the same period $T$.\nPeriodic solutions model rhythms such as oscillating circuits, heartbeats and planetary motion.\n\n## Explanations\n\n### L1 human\n\nA periodic solution is a motion that comes back to exactly the same state after a fixed time and then repeats forever.\nIts graph is a [[periodic function|0ATF6Q0633]] of time.\nThe function $x(t) = \\cos t$ solves $x'' + x = 0$ and repeats after the time $2\\pi$.\nA pendulum that swings back and forth without friction moves along a periodic solution.\nA damped pendulum does not, because its swings get smaller and smaller.\n",
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