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  "body": "\n## Statement\n\nLet $U$ be an open subset of $\\R^n$, let $k$ be an integer with $1 \\le k \u003c \\infty$, and let $f \\colon U \\to \\R^n$ be of class $C^k$.\nLet $x_0 \\in U$ be an [[equilibrium point|0G4X36AE0X]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.\nLet $E^c$ be the sum of the generalised eigenspaces of the [[Jacobian matrix|047NDTEPCV]] $Df(x_0)$ for the eigenvalues with real part $0$.\nThen $x_0$ has a [[local center manifold|F35WJP8VHG]] $W^c$ of class $C^k$.\nThat is, $W^c$ is an embedded $C^k$ submanifold of a neighbourhood of $x_0$ that contains $x_0$.\nIts tangent space at $x_0$ is $E^c$, and it is locally [[invariant|WWZTW11664]] under the [[flow|02903SSA48]] of $f$.\n\n## Remarks\n\nThe usual proof first cuts $f$ off outside a small ball, so that the nonlinear part is globally small in $C^1$.\nFor the modified equation, a global center manifold is the graph of a map $E^c \\to E^s \\oplus E^u$.\nThis map is found as a fixed point of a contraction on a space of functions with controlled growth; this is the Lyapunov–Perron method.\nThe center manifold need not be unique.\nFor $x' = x^2$, $y' = -y$, every curve made of $y = c e^{1/x}$ for $x \u003c 0$ and $y = 0$ for $x \\ge 0$ is a center manifold of the origin.\nThe finiteness of $k$ matters.\nFor a $C^\\infty$ system there is a $C^k$ center manifold for each finite $k$, but in general no $C^\\infty$ one, and the neighbourhood may shrink as $k$ grows.\nIf no eigenvalue of $Df(x_0)$ has positive real part, the stability of $x_0$ is the same as for the restriction of the equation to $W^c$.\n\n## Explanations\n\n### L1 human\n\nAt a rest point, the linear part of the equation splits the directions into contracting, expanding and neutral ones.\nIn the contracting and expanding directions the linear part already tells what happens.\nThe theorem says that the neutral directions bend into a curved invariant surface through the rest point.\nAll the delicate behaviour happens on this surface, which has fewer dimensions than the whole space.\n\n### L2 human\n\nThe theorem is the main tool for studying non-hyperbolic equilibria, where the [[Hartman–Grobman theorem|0S8WX8A4V9]] gives no information.\nOne computes a Taylor expansion of the center manifold as a graph over $E^c$ by matching coefficients in the invariance equation.\nThen one studies the reduced equation on $E^c$, which in a bifurcation is often one- or two-dimensional.\nThis reduction underlies the normal forms of the saddle-node and Hopf bifurcations.\nIt complements the [[stable manifold theorem|WF22KK7C3R]], which treats the contracting and expanding directions.\n",
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