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  "body": "\n## Statement\n\nLet $U$ be an open subset of $\\R^n$, let $k \\ge 1$, and let $f \\colon U \\to \\R^n$ be of class $C^k$.\nLet $x_0 \\in U$ be a [[hyperbolic equilibrium point|01TPMXM6NM]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$, with [[flow|02903SSA48]] $\\varphi$.\nLet $A = Df(x_0)$ be the [[Jacobian matrix|047NDTEPCV]].\nLet $E^s$ be the sum of the generalised eigenspaces of $A$ for eigenvalues with negative real part.\nLet $E^u$ be the sum of those for eigenvalues with positive real part.\nThen there is an open neighbourhood $V$ of $x_0$ in $U$ such that:\n\n1. The local stable manifold $W^s_{\\mathrm{loc}} = \\set{x \\in V : \\varphi_t(x) \\in V \\text{ for all } t \\ge 0}$ is an embedded $C^k$ submanifold of $\\R^n$ of dimension $\\dim E^s$ whose tangent space at $x_0$ is $E^s$.\n2. For each $x \\in W^s_{\\mathrm{loc}}$, $\\varphi_t(x) \\to x_0$ as $t \\to +\\infty$.\n3. The local unstable manifold $W^u_{\\mathrm{loc}} = \\set{x \\in V : \\varphi_t(x) \\in V \\text{ for all } t \\le 0}$ is an embedded $C^k$ submanifold of dimension $\\dim E^u$ with tangent space $E^u$ at $x_0$.\n4. For each $x \\in W^u_{\\mathrm{loc}}$, $\\varphi_t(x) \\to x_0$ as $t \\to -\\infty$.\n\n## Remarks\n\nOne proof writes the solutions that stay near $x_0$ as fixed points of an integral equation built from the variation of constants formula.\nThen it applies the contraction mapping principle; another proof uses the graph transform.\nThe global stable manifold $W^s(x_0) = \\set{x \\in U : \\varphi_t(x) \\to x_0 \\text{ as } t \\to \\infty}$ is the union of the sets $\\varphi_{-t}(W^s_{\\mathrm{loc}})$ for $t \\ge 0$, and it is an [[invariant set|WWZTW11664]].\nExample: for $x' = -x$, $y' = y + x^2$, the origin is hyperbolic with $E^s$ the $x$-axis and $E^u$ the $y$-axis.\nOn the curve $y = -x^2/3$ one has $y' = (-2x/3)(-x) = 2x^2/3 = y + x^2$, so the curve is invariant; it is the stable manifold, tangent to $E^s$, while the unstable manifold is the $y$-axis.\nHyperbolicity is needed: for $x' = x^2$, $y' = -y$, the set of points that tend to the origin is the closed half-plane $x \\le 0$, which is not a curve.\n\n## Explanations\n\n### L1 human\n\nNear a saddle point, the points that flow into the saddle form a smooth surface, and the points that come out of it form another.\nThese surfaces are curved copies of the flat stable and unstable directions of the linearised system.\n\n### L2 human\n\nThe theorem is the nonlinear version of the splitting of $\\R^n$ into $E^s \\oplus E^u$ for a linear system.\nTogether with the [[Hartman–Grobman theorem|0S8WX8A4V9]], it gives the local picture near a hyperbolic equilibrium.\nStable and unstable manifolds organise the global phase portrait: their intersections give homoclinic and heteroclinic orbits.\nThe same result holds for hyperbolic fixed points of diffeomorphisms and for hyperbolic periodic orbits.\n",
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