The Stable Manifold Theorem Applications

Question

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a $k$-dimensional differentiable manifold tangent to the stable subspace $E^s$ of the linear system $x'=Ax$ where $A=Df(0)$ and $0$ is a hyperbolic equilibrium point. Show that $W^s(0)$ is unique and invariant with respect to the flow $\phi_t(x)$ ; furthemore, for all $x\in W^s(0)$, $$\lim_{t \rightarrow \infty} \phi_t(x)=0.$$

Answer 1

If $S \subset W^s(0)$ and $S$ contains a $W^s$ neighborhood of $0$, then $$ W^s(0) = \bigcup_{t \leq 0} \phi^t(S) \, , ~~~~~~ (*) $$ where above we have set $$ W^s(0) = \{ x \in \mathbb R^n : \lim_{t \to \infty} \phi^t(x) = 0\} \, . $$

In this context, $S$ is often referred to as a local stable manifold at $0$. The following is a statement of the "local" version of the stable manifold theorem. Below we write $B(r)$ for the ball of radius $r$ centered at the origin.

Local stable manifold theorem. Let $\epsilon > 0$ be sufficiently small. Then, $W^s_{loc}(0) = \{ x \in B(\epsilon) : \phi^t(x) \in B(\epsilon) \text{ for all } t \geq 0 \}$ is a smooth manifold (as smooth as $\phi^t$) tangent to $E^s$ at the origin, and for any $x \in W^s_{loc}(0)$, we have $\lim_{t \to 0} \phi^t(x) = 0$.

This is proved in many books. See, e.g., Vaughn Climenhaga's blog post for the proof in the case of hyperbolic fixed points of ODE.

Once it is known, it is straightforward to prove the identity ($*$). Assume $S = W^s_{loc}(0)$ for some $\epsilon > 0$. The containment "$\supset$" follows on noting that if $x \in \phi^{-s}(S), s \geq 0$, then $\phi^{s} x \in S$ and so $\lim_{t \to \infty} \phi^t(x) = \lim_{t \to \infty} \phi^{t - s} (\phi^{s} x) = 0$ by the local stable manifold theorem. For the containment "$\subset$", let $x \in W^s(0)$. Then there exists $T$ such that $\phi^t(x) \in B(\epsilon)$ for all $t \geq T$. Setting $x' = \phi^T(x)$, it follows from the local stable manifold theorem that $x' \in W_{loc}^s(0) = S$, hence $x \in \phi^{-T} S$.

Answer 2

Both properties are immediate from the definitions. Note that $S$ is unique is a sufficiently small neighborhood and so $W^s(0)$ is well defined, that is, the set $W^s(0)$ is independent of $S$. It is also invariant since $\phi_t(S)\subset S$ for all positive $t$.

The last property follows from the definition: indeed, for each $x\in W^s(0)$ there exists $t>0$ such that $\phi_t(x)\in S$ (by the definition of $S$) and so $\phi_s(x)=\phi_{s-t}(\phi_t(x))\to 0$ when $s\to+\infty$ (since this happens if and only if $t-s\to+\infty$.