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  "body": "\n## Statement\n\nLet $r \\ge 1$ be an integer, and let $f \\colon \\R^n \\times \\R^m \\times \\R \\to \\R^n$ and $g \\colon \\R^n \\times \\R^m \\times \\R \\to \\R^m$ be of class $C^{r+1}$.\nConsider the [[singularly perturbed|G6J21TGRMM]] slow-fast system of [[ordinary differential equations|0NT64JV96A]]\n$$\\dot x = f(x, y, \\varepsilon), \\quad \\varepsilon \\dot y = g(x, y, \\varepsilon).$$\nLet $K \\subseteq \\R^n$ be a compact set that is the closure of an open set with smooth boundary.\nLet $h_0 \\colon K \\to \\R^m$ be of class $C^{r+1}$ with $g(x, h_0(x), 0) = 0$ for all $x \\in K$.\nSo $M_0 = \\set{(x, h_0(x)) : x \\in K}$ is part of the *critical manifold* $\\set{g = 0}$.\nAssume that $M_0$ is *normally hyperbolic*: for each $x \\in K$, no [[eigenvalue|0BQAA2BK3Q]] of $\\partial_y g(x, h_0(x), 0)$ has real part $0$.\nThen there is $\\varepsilon_1 \u003e 0$ such that for $0 \u003c \\varepsilon \\le \\varepsilon_1$ there is a $C^r$ function $h_\\varepsilon \\colon K \\to \\R^m$ with these properties.\n\n1. The *slow manifold* $M_\\varepsilon = \\set{(x, h_\\varepsilon(x)) : x \\in K}$ is locally [[invariant|WWZTW11664]].\n   A solution that starts on $M_\\varepsilon$ stays on it as long as $x$ remains in the interior of $K$.\n2. $h_\\varepsilon$ depends in a $C^r$ way on $\\varepsilon$, and $h_\\varepsilon = h_0 + O(\\varepsilon)$ uniformly on $K$.\n3. Suppose that along $M_0$, the matrix $\\partial_y g$ has $k$ eigenvalues with negative and $m - k$ with positive real part.\n   Then $M_\\varepsilon$ has local stable and unstable manifolds of dimensions $n + k$ and $n + m - k$.\n   They are $O(\\varepsilon)$-close to the stable and unstable manifolds of $M_0$ for the fast system at $\\varepsilon = 0$.\n\n## Remarks\n\nAt $\\varepsilon = 0$, every point of $M_0$ is an equilibrium of the fast system $y' = g(x, y, 0)$, where the prime is the derivative in $\\tau = t / \\varepsilon$.\nNormal hyperbolicity means that the contraction or expansion transverse to $M_0$ dominates the dynamics along $M_0$.\nThe proof applies the persistence theory of normally hyperbolic invariant manifolds, developed by Fenichel, to the system in the fast time with $\\varepsilon$ added as a variable.\n\nExample: for $\\dot x = -x$, $\\varepsilon \\dot y = -y + x^2$, the critical manifold is $y = x^2$, normally attracting since $\\partial_y g = -1$.\nOne checks that $y = h_\\varepsilon(x) = x^2 / (1 - 2\\varepsilon)$ is invariant for $\\varepsilon \u003c 1/2$: along it $\\varepsilon \\dot y = -2\\varepsilon x^2 / (1 - 2\\varepsilon) = -y + x^2$.\nSo $h_\\varepsilon = x^2 + O(\\varepsilon)$, as the theorem predicts.\n\nNormal hyperbolicity is needed: at fold points, where an eigenvalue of $\\partial_y g$ crosses $0$, the slow manifold can fail to continue, as in relaxation oscillations and canard phenomena.\nThe slow manifold is in general not unique, but any two choices are exponentially close in $1/\\varepsilon$.\n\n## Explanations\n\n### L1 human\n\nWhen a system has fast and slow parts, the fast part quickly settles near a surface of rest points.\nFenichel's theorem says that, for a small but positive speed ratio, there is a true surface nearby that the motion follows exactly.\nOn that surface, the system can be studied with fewer variables.\n\n### L2 human\n\nThe theorem is the foundation of geometric singular perturbation theory.\nIt complements [[Tikhonov's theorem|8X8Y61MGHW]], which gives convergence on finite time intervals.\nOn the slow manifold, the flow is a regular perturbation of the reduced flow $\\dot x = f(x, h_0(x), 0)$.\nCombined with the stable and unstable fibrations, it is used to construct travelling waves and periodic orbits of relaxation type.\n",
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