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  "body": "\n## Definition\n\nLet $D \\subseteq \\R^n$ be open, let $T \u003e 0$, and let $f \\colon D \\times \\R \\to \\R^n$ be [[continuously differentiable|0VW90BBSJ4]] and [[periodic|0ATF6Q0633]] of period $T$ in its last variable.\nFor a small parameter $\\varepsilon \u003e 0$, consider the [[differential equation|0NT64JV96A]] in standard form\n$$x' = \\varepsilon f(x, t).$$\nThe *averaged function* is\n$$\\bar f(y) = \\frac{1}{T} \\int_0^T f(y, t)\\, dt, \\qquad y \\in D,$$\nand the *averaged equation* is $y' = \\varepsilon \\bar f(y)$.\n*Krylov–Bogolyubov averaging* is the method of approximating the solutions of the equation in standard form by the solutions of the averaged equation.\n\n## Remarks\n\nThe first averaging theorem justifies the method.\nLet $x$ and $y$ solve the two equations with $x(0) = y(0)$, and let $L \u003e 0$ be such that $y(t)$ stays in a compact subset of $D$ for $0 \\le t \\le L/\\varepsilon$.\nThen there is a constant $C$, independent of $\\varepsilon$, with $\\norm{x(t) - y(t)} \\le C \\varepsilon$ for $0 \\le t \\le L/\\varepsilon$ and all small $\\varepsilon$.\n\nExample: $f(x, t) = x \\sin^2 t$ on $\\R$ with $T = \\pi$.\nThen $\\bar f(y) = \\frac{1}{\\pi} \\int_0^\\pi y \\sin^2 t\\, dt = \\frac{1}{2} y$.\nThe solutions are $x(t) = x_0 e^{\\varepsilon (t/2 - \\sin(2t)/4)}$ and $y(t) = x_0 e^{\\varepsilon t/2}$.\nTheir ratio is $e^{-\\varepsilon \\sin(2t)/4} = 1 + O(\\varepsilon)$, so the difference is $O(\\varepsilon)$ on intervals of length of order $1/\\varepsilon$.\n\nNon-example: $x' = \\varepsilon x t$ is not periodic in $t$, so it is not in the standard form above, and there is no period over which to average.\n\nWeakly nonlinear oscillators $u'' + u = \\varepsilon h(u, u')$ are put in standard form by writing $u = r \\cos(t + \\phi)$, $u' = -r \\sin(t + \\phi)$ and using $(r, \\phi)$ as the new unknowns.\n",
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