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stable manifold theorem (Theorem)

Let $ E$ be an open subset of $ \mathbb{R}^n$ containing the origin, let $ f\in C^1(E)$, and let $ \phi_t$ be the flow of the nonlinear system $ x'=f(x)$.

Suppose that $ f(x_0)=0$ and that $ Df(x_0)$ has $ k$ eigenvalues with negative real part and $ n-k$ eigenvalues with positive real part. Then there exists a $ k$-dimensional differentiable manifold $ S$ tangent to the stable subspace $ E^S$ of the linear system $ x'=Df(x)x$ at $ x_0$ such that for all $ t\geq 0$, $ \phi_t(S)\subset S$ and for all $ y\in S$,

$\displaystyle \lim_{t\to\infty}\phi_t(y)=x_0 $
and there exists an $ n-k$ dimensional differentiable manifold $ U$ tangent to the unstable subspace $ E^U$ of $ x'=Df(x)x$ at $ x_0$ such that for all $ t\leq 0$, $ \phi_t(U)\subset U$ and for all $ y\in U$,
$\displaystyle \lim_{t\to -\infty}\phi_t(y)=x_0. $



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Cross-references: subspace, unstable, linear system, stable subspace, tangent, differentiable manifold, positive, real part, negative, eigenvalues, system, flow, origin, open subset
There is 1 reference to this entry.

This is version 1 of stable manifold theorem, born on 2002-08-18.
Object id is 3315, canonical name is StableManifoldTheorem.
Accessed 3557 times total.

Classification:
AMS MSC: 34C99 (Ordinary differential equations :: Qualitative theory :: Miscellaneous)
Pending Errata and Addenda
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Discussion
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vis a vis hyperbolic set by Linas on 2006-06-09 12:30:52
How does this article differ from that for the hyperbolic set,

http://planetmath.org/?op=getobj&from=objects&id=4338

should that article and this article be merged?
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