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  "body": "\n## Summary\n\nA center manifold of an [[equilibrium point|0N8NGKDZDZ]] is an [[invariant|WWZTW11664]] manifold through the equilibrium.\nIt is tangent to the directions where the linearisation neither grows nor decays.\nThe center manifold theorem says that such a manifold exists, though it need not be unique.\nFor an equation of class $C^k$ with finite $k$, it can be taken of class $C^k$.\nThe interesting dynamics near a non-hyperbolic equilibrium happens on the center manifold.\nSo the study of stability and bifurcations reduces to a system of lower dimension.\n\n## Explanations\n\n### L1 human\n\nAt an equilibrium, some directions pull solutions in fast, some push them out fast, and some do neither at the linear level.\nThe neutral directions are where the nonlinear terms decide what happens.\nThe center manifold is a curved surface along these neutral directions that solutions near the equilibrium follow.\nFor $x' = x^2$, $y' = -y$, the $y$-direction contracts and the $x$-axis is a center manifold, on which the motion is $x' = x^2$.\nSo all the delicate behaviour can be read from a one-dimensional equation.\n",
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