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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$ and let $f : U \\to \\mathbb{R}^n$ be locally Lipschitz.\nFor $x_0 \\in U$, let $\\gamma_{x_0} : I(x_0) \\to U$ be the [[maximal solution|019VSXSFMQ]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ with $x(0) = x_0$.\nLet $\\Omega = \\{(t, x_0) : x_0 \\in U,\\ t \\in I(x_0)\\}$.\nThe flow of $f$ is the map $\\varphi : \\Omega \\to U$ given by $\\varphi(t, x_0) = \\gamma_{x_0}(t)$.\nAnother notation is $\\varphi_t(x_0) = \\varphi(t, x_0)$.\n\n## Remarks\n\nThe flow has the group properties $\\varphi_0(x) = x$ and $\\varphi_s(\\varphi_t(x)) = \\varphi_{s + t}(x)$, whenever both sides are defined. The second property follows from [[uniqueness|025PKW8AMR]] and [[time translation|0WA7ZR95V4]].\nThe set $\\Omega$ is open and $\\varphi$ is continuous on $\\Omega$.\nIf $\\Omega = \\mathbb{R} \\times U$, the flow is complete. Then $t \\mapsto \\varphi_t$ is an action of the additive group $\\mathbb{R}$ on $U$.\nFor a linear system $x' = A x$, the flow is $\\varphi_t(x) = e^{tA} x$, the [[matrix exponential|0YC8JT513J]].\n\n## Explanations\n\n### L1 human\n\nThe flow moves each point of the phase space along the solution through it.\nThe map $\\varphi_t$ says where each point is after time $t$.\nMoving for time $t$ and then for time $s$ is the same as moving for time $s + t$.\n",
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