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  "body": "\n## Statement\n\nLet $f \\colon \\R^n \\times \\R^m \\to \\R^n$ and $g \\colon \\R^n \\times \\R^m \\to \\R^m$ be continuously differentiable.\nFor $\\varepsilon \u003e 0$ consider the [[singularly perturbed|G6J21TGRMM]] [[initial value problem|0FXDW992DS]] for a system of [[ordinary differential equations|0NT64JV96A]]\n$$\\dot x = f(x, y), \\quad \\varepsilon \\dot y = g(x, y), \\quad x(0) = x^0, \\quad y(0) = y^0.$$\nLet $T \u003e 0$, and assume the following.\n\n1. There are an open set $U \\subseteq \\R^n$ and a continuously differentiable $\\varphi \\colon U \\to \\R^m$ with $g(x, \\varphi(x)) = 0$ for all $x \\in U$.\n2. For each $x \\in U$, all eigenvalues of the matrix $\\partial_y g(x, \\varphi(x))$ have negative real part.\n3. The *reduced problem* $\\dot{\\bar x} = f(\\bar x, \\varphi(\\bar x))$, $\\bar x(0) = x^0$, has a solution $\\bar x$ on $[0, T]$ with values in $U$.\n4. The solution of the *boundary layer problem* $\\frac{d\\hat y}{d\\tau} = g(x^0, \\hat y)$, $\\hat y(0) = y^0$, exists for all $\\tau \\ge 0$ and tends to $\\varphi(x^0)$ as $\\tau \\to \\infty$.\n\nThen there is $\\varepsilon_1 \u003e 0$ such that for $0 \u003c \\varepsilon \u003c \\varepsilon_1$ the solution $(x_\\varepsilon, y_\\varepsilon)$ exists on $[0, T]$.\nMoreover, as $\\varepsilon \\to 0$, $x_\\varepsilon \\to \\bar x$ uniformly on $[0, T]$, and $y_\\varepsilon \\to \\varphi(\\bar x)$ uniformly on $[t_0, T]$ for every $t_0 \\in (0, T]$.\n\n## Remarks\n\nIn the fast time $\\tau = t / \\varepsilon$, the variable $x$ hardly moves, and $y$ follows the boundary layer equation, in which $\\varphi(x)$ is an [[asymptotically stable|0NB689782W]] [[equilibrium point|0N8NGKDZDZ]] by hypothesis 2.\nSo $y$ first jumps quickly to near the slow manifold $y = \\varphi(x)$, and then the system moves slowly along it according to the reduced problem.\nThe proof turns this into estimates with a Lyapunov function for the fast subsystem, uniform in $x$ on a compact neighbourhood of the reduced trajectory.\n\nExample: $\\dot x = -x + y$, $\\varepsilon \\dot y = x - 2y$, with $x(0) = 1$, $y(0) = 5$.\nHere $\\varphi(x) = x / 2$ and $\\partial_y g = -2 \u003c 0$, and the reduced problem $\\dot{\\bar x} = -\\bar x / 2$ gives $\\bar x(t) = e^{-t/2}$.\nThe theorem gives $x_\\varepsilon(t) \\to e^{-t/2}$ and $y_\\varepsilon(t) \\to e^{-t/2} / 2$ for $t \u003e 0$, while $y_\\varepsilon(0) = 5$, so $y$ has a [[boundary layer|YCFPD0GE5G]] at $t = 0$.\n\nStability of the slow manifold is needed.\nFor $\\dot x = 0$, $\\varepsilon \\dot y = y$ with $y(0) = y^0 \\neq 0$, the root is $\\varphi = 0$ but $\\partial_y g = 1 \u003e 0$, and $y_\\varepsilon(t) = y^0 e^{t/\\varepsilon}$ does not converge.\n\n## Explanations\n\n### L1 human\n\nSome systems have a fast part and a slow part.\nIf the fast part always settles quickly to a rest state, one can replace it by that rest state and only study the slow part.\nThe theorem says this replacement gives the right answer as the speed gap becomes infinite, except for a short initial jump.\n\n### L2 human\n\nTikhonov's theorem justifies the quasi-steady-state approximation of chemical kinetics, such as the Michaelis–Menten law in enzyme kinetics.\nIt is a result on finite time intervals; Fenichel's theorem, the basis of geometric singular perturbation theory, gives the invariant slow manifold for $\\varepsilon \u003e 0$.\nThe approximation can be refined to all orders by the method of matched asymptotic expansions, under extra smoothness.\n",
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