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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be locally Lipschitz, and consider the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.\nA [[solution|04SW7WECS8]] $\\gamma : \\mathbb{R} \\to U$ is periodic if it is not constant and there is a real number $T \u003e 0$ with $\\gamma(t + T) = \\gamma(t)$ for each $t \\in \\mathbb{R}$.\nThe smallest such $T$ is the period of $\\gamma$.\nA periodic orbit, or closed orbit, is the [[trajectory|0BVT6VSB3S]] $\\gamma(\\mathbb{R})$ of a periodic solution $\\gamma$.\n\n## Remarks\n\nA non-constant periodic solution has a smallest period $T \u003e 0$, because $\\gamma$ is continuous and not constant.\nA periodic orbit is a simple closed curve in $U$.\nLet a maximal solution $\\gamma$ that is not constant satisfy $\\gamma(t_1 + T) = \\gamma(t_1)$ for one $t_1$ and one $T \u003e 0$. Then, by [[uniqueness|025PKW8AMR]], $\\gamma$ is defined on $\\mathbb{R}$ and is periodic.\nFor the [[harmonic oscillator|0YG0ZJPTWG]] $x' = y$, $y' = -x$, each circle $x^2 + y^2 = r^2$ with $r \u003e 0$ is a periodic orbit of period $2\\pi$.\n\n## Explanations\n\n### L1 human\n\nA periodic orbit is a motion that repeats itself exactly after a fixed time.\nIn the phase space it is a closed loop that the state goes around again and again.\n",
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