Ordinary differential equations Qualitative theory Theorem

Stable manifold theorem

Builds on Autonomous system Flow of a vector field Hyperbolic equilibrium point Jacobian matrix Invariant set

Statement

Let $U$ be an open subset of $\R^n$ , let $k \ge 1$ , and let $f \colon U \to \R^n$ be of class $C^k$ . Let $x_0 \in U$ be a hyperbolic equilibrium point of the autonomous system $x' = f(x)$ , with flow $\varphi$ . Let $A = Df(x_0)$ be the Jacobian matrix. Let $E^s$ be the sum of the generalised eigenspaces of $A$ for eigenvalues with negative real part. Let $E^u$ be the sum of those for eigenvalues with positive real part. Then there is an open neighbourhood $V$ of $x_0$ in $U$ such that:

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 $E^s$ .
For each $x \in W^s_{\mathrm{loc}}$ , $\varphi_t(x) \to x_0$ as $t \to +\infty$ .
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 $x_0$ .
For each $x \in W^u_{\mathrm{loc}}$ , $\varphi_t(x) \to x_0$ as $t \to -\infty$ .

Remarks

One proof writes the solutions that stay near $x_0$ as fixed points of an integral equation built from the variation of constants formula. Then it applies the contraction mapping principle; another proof uses the graph transform. The 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. Example: for $x' = -x$ , $y' = y + x^2$ , the origin is hyperbolic with $E^s$ the $x$ -axis and $E^u$ the $y$ -axis. On 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. Hyperbolicity 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.

Going deeper

For a student

The theorem is the nonlinear version of the splitting of $\R^n$ into $E^s \oplus E^u$ for a linear system. Together with the Hartman–Grobman theorem, it gives the local picture near a hyperbolic equilibrium. Stable and unstable manifolds organise the global phase portrait: their intersections give homoclinic and heteroclinic orbits. The same result holds for hyperbolic fixed points of diffeomorphisms and for hyperbolic periodic orbits.

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Differential equation concept
Euclidean space object
Function concept
Hyperbolic equilibrium point concept
Interval concept
Invariant set concept
Ordinary differential equation definition
Total derivative concept
Function of several variables concept
Lipschitz condition definition
Grönwall's inequality statement
Partial derivative concept
Solution of an ordinary differential equation definition
Autonomous system definition
Initial value problem definition
Jacobian matrix concept
Uniqueness of solutions for Lipschitz vector fields statement
Maximal solution definition
Flow of a vector field definition
Stable manifold theorem statement

Tag	`WF22KK7C3R`
URI	`m2://WF22KK7C3R`
Kind	statement, theorem
Writing	W0 stub
MSC	34C45, 37D10
Status	This copy of the site has no build data. Run `m2 status WF22KK7C3R`.
About	Hyperbolic equilibrium point, Invariant set
Uses	Autonomous system, Flow of a vector field, Hyperbolic equilibrium point, Jacobian matrix, Invariant set
Same as	Wikidata Q7595768 Wikipedia Stable manifold theorem
Changed	2 November 2026