# stable manifold 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 $x^{\prime}=f(x)$.

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 $x^{\prime}=Df(x)x$ at $x_{0}$ such that for all $t\geq 0$, $\phi_{t}(S)\subset S$ and for all $y\in S$,

 $\lim_{t\to\infty}\phi_{t}(y)=x_{0}$

and there exists an $n-k$ dimensional differentiable manifold $U$ tangent to the unstable subspace  $E^{U}$ of $x^{\prime}=Df(x)x$ at $x_{0}$ such that for all $t\leq 0$, $\phi_{t}(U)\subset U$ and for all $y\in U$,

 $\lim_{t\to-\infty}\phi_{t}(y)=x_{0}.$
Title stable manifold theorem StableManifoldTheorem 2013-03-22 12:57:17 2013-03-22 12:57:17 jarino (552) jarino (552) 4 jarino (552) Theorem msc 34C99