stable manifold theorem

Let $E$ be an open subset of ${ℝ}^n$ containing the origin, let$f\in C^1(E)$, and let ${\varphi }_t$ 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 $t\ge 0$, ${\varphi }_t(S)\subset S$ and for all $y\in S$,

$$\underset{t\to \mathrm{\infty }}{lim}{\varphi }_t(y)=x_0$$

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$t\le 0$, ${\varphi }_t(U)\subset U$ and for all $y\in U$,

$$\underset{t\to -\mathrm{\infty }}{lim}{\varphi }_t(y)=x_0.$$