Stable Manifold Theorem

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Mathematics

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 $y\in S$, \[ \lim_{t\to\infty}\phi_t(y)=x_0 \] 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 $y\in U$, \[ \lim_{t\to -\infty}\phi_t(y)=x_0. \]