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  "body": "\n## Definition\n\nSeveral formal models of hybrid systems are in use; a common one is the hybrid automaton.\nA *hybrid automaton* with continuous state space $\\R^n$ consists of the following data.\n\n1. A finite set $Q$ of discrete states, called modes.\n2. For each mode $q \\in Q$, a [[vector field|0299VCW1TH]] $f_q \\colon \\R^n \\to \\R^n$ and a set $D_q \\subseteq \\R^n$, the domain of $q$.\n3. A set $E \\subseteq Q \\times Q$ of edges, that is, allowed mode switches.\n4. For each edge $e \\in E$, a guard $G_e \\subseteq \\R^n$ and a reset map $R_e$ that gives a nonempty set $R_e(x) \\subseteq \\R^n$ for each $x \\in G_e$.\n5. A set $\\mathrm{Init} \\subseteq Q \\times \\R^n$ of initial states.\n\nAn *execution* is a finite or infinite sequence of intervals $[\\tau_0, \\tau_0'], [\\tau_1, \\tau_1'], \\ldots$ with $\\tau_i' = \\tau_{i+1}$.\nOn the interval $[\\tau_i, \\tau_i']$ it has a mode $q_i$ and a curve $x_i$ that solves $x' = f_{q_i}(x)$ and stays in $D_{q_i}$.\nThe start satisfies $(q_0, x_0(\\tau_0)) \\in \\mathrm{Init}$.\nAt each switch, $e = (q_i, q_{i+1}) \\in E$, $x_i(\\tau_i') \\in G_e$ and $x_{i+1}(\\tau_{i+1}) \\in R_e(x_i(\\tau_i'))$.\nA *hybrid system* is a system modelled by such data, or by an equivalent formalism with flows and jumps.\n\n## Remarks\n\nThe discrete part alone is a finite-state machine, and the continuous part in one mode is an [[ordinary differential equation|0NT64JV96A]].\nExample: the bouncing ball has one mode and the state $x = (h, v)$ of height and velocity.\nIts vector field is $f(h, v) = (v, -g)$ with $g \u003e 0$, and its domain is $h \\ge 0$.\nIt has one edge from the mode to itself, with guard $\\set{h = 0, v \\le 0}$ and reset $R(0, v) = \\set{(0, -c v)}$, where $0 \\le c \u003c 1$.\nBetween impacts the ball follows a parabola, and each impact multiplies its speed by $c$.\nFor $0 \u003c c \u003c 1$ the impacts accumulate at a finite time; this Zeno behaviour has no analogue for ordinary differential equations.\nNon-example: data with infinitely many modes, one for each integer, fail clause 1, which asks for a finite set of modes; such models need a more general framework.\n",
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