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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|WBZHXGKNSJ]] and let $r \\ge 1$ be an integer or $\\infty$.\nA *smooth dynamical system of class $C^r$* on $M$ is one of the following.\n\n1. Discrete time: a $C^r$ [[diffeomorphism|0Z1Y7JHGBR]] $f \\colon M \\to M$, with the action $n \\mapsto f^n$ of $\\Z$ on $M$.\n2. Continuous time: a $C^r$ map $\\varphi \\colon \\R \\times M \\to M$, written $\\varphi_t(x) = \\varphi(t, x)$, such that $\\varphi_0 = \\id_M$ and $\\varphi_{s+t} = \\varphi_s \\circ \\varphi_t$ for all $s, t \\in \\R$.\n\n## Remarks\n\nIn case 2 each $\\varphi_t$ is a $C^r$ diffeomorphism with inverse $\\varphi_{-t}$, and $\\varphi$ is called a $C^r$ flow.\nIt is the flow of the vector field $X(x) = \\frac{d}{dt}\\varphi_t(x)\\big|_{t=0}$, which is of class $C^{r-1}$.\nConversely, on a compact manifold every $C^r$ vector field generates a complete $C^r$ flow.\nSome authors also allow non-invertible $C^r$ maps $f \\colon M \\to M$ in case 1.\nExample: on the circle $M = \\R/\\Z$, the rotation $f(x) = x + \\alpha \\bmod 1$ is a $C^\\infty$ diffeomorphism, and $\\varphi_t(x) = x + t \\bmod 1$ is a $C^\\infty$ flow.\nNon-example: $f(x) = 2x \\bmod 1$ on the circle is smooth but two-to-one, so it is not a diffeomorphism and fails case 1 in this definition.\n",
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