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  "body": "\n## Definition\n\nLet $\\mathbb{T}$ be an additive [[monoid|0N6R902FG1]], the *time*, usually $\\N$, $\\Z$, $[0, \\infty)$ or $\\R$.\nLet $X$ be a [[set|0BSJ7RQ9BW]], the *phase space* or *state space*.\nA *dynamical system* with time $\\mathbb{T}$ on $X$ is a [[map|0EPAT51DXQ]] $\\Phi \\colon \\mathbb{T} \\times X \\to X$ such that:\n\n1. $\\Phi(0, x) = x$ for every $x \\in X$;\n2. $\\Phi(s, \\Phi(t, x)) = \\Phi(s + t, x)$ for all $s, t \\in \\mathbb{T}$ and $x \\in X$.\n\nThe map $\\Phi^t = \\Phi(t, \\cdot) \\colon X \\to X$ is the *time-$t$ map*, and the set $\\set{\\Phi(t, x) : t \\in \\mathbb{T}}$ is the *orbit* of $x$.\n\n## Remarks\n\nClauses 1 and 2 say that $t \\mapsto \\Phi^t$ is a monoid homomorphism from $\\mathbb{T}$ to the maps $X \\to X$ under composition, that is, an action of $\\mathbb{T}$ on $X$.\nIn practice $X$ carries more structure, and $\\Phi$ is required to respect it.\nExamples are continuous maps on a topological space, smooth maps on a manifold, or measurable maps on a measure space.\nWith $\\mathbb{T} = \\R$ the system is a flow, with $\\mathbb{T} = [0, \\infty)$ a semiflow, and with $\\mathbb{T} = \\N$ or $\\Z$ a discrete-time system.\n\nExample: $\\mathbb{T} = \\R$, $X = \\R$ and $\\Phi(t, x) = e^{-t} x$, the solutions of $x' = -x$.\nClause 1 holds since $e^{0} x = x$, and clause 2 holds since $e^{-s}(e^{-t} x) = e^{-(s + t)} x$.\n\nNon-example: $\\Phi(t, x) = x + t^2$ on $\\R$ with $\\mathbb{T} = \\R$.\nClause 1 holds, but $\\Phi(1, \\Phi(1, 0)) = 2$ while $\\Phi(2, 0) = 4$, so clause 2 fails.\n",
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