Stable manifold theorem

Question

The stable manifold theorem says that there exists a stable manifold and an unstable manifold with the dimension corresponding to the dimension of its eigenspace of its linearized operator. I have the following question

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1. Is such decomposition unique, i.e. Can we find other stable manifold?
2. What about the behavior of the point not on these two manifold? By Hartman Gorbman, we know if a point is on stable manifold , then at some time t<0, it will go out a neighborhood of the equilibrium point