In a given context of a couple of ODE's, it's proved that the local stable manifold through one equilibrium point is a vertical segment, once we have by calculus that it's tangent to $[0,1]$ and has dimension $1$.
I was thinking if there's some result that allows us to extend local to global stable manifold and say that the global stable manifold through this equilibrium point is the vertical line (of $\mathbb{R}^2$) through it. Is it possible?
If not, could you give some counterexample or explain why?
Many thanks.
EDIT: Once the question seems to vague, please, could you so reference some good (and relatively simple - I mean, MsC Level) for extend results to local stable manifolds to global stable manifolds?