# What is a strong stable manifold?

For a dynamical system in $\mathbb{R}^n$ given by $\dot{x} = f(x)$, and a fixed point $p$, one defines stable and unstable manifolds at the point $p$. These are well documented, and a quick Google/Wikipedia search provides all the information one needs (and more).

There is, however, also the notion of strong stable manifold, and I have been so far unable to find what it precisely means. Could someone be so kind as to either provide the definition, or a reference to literature where it is defined?

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As an example, for the system $(\dot{x}_1,\dot{x}_2) = (-x_1,-10 x_2)$, $p=0$ is a fixed point. The relevant eigenvalues are -1 and -10, which are both negative, so the whole plane is the stable manifold. The corresponding eigenvectors are $(1,0)$ and $(0,1)$ respectively. The exact solution is $(x_1(t),x_2(t)) = (x_1(0) {\rm e}^{-t},x_2(0) {\rm e}^{-10 t})$. Since the latter exponential decays much faster than the first one, the solution rapidly approaches the line through the origin with direction $(1,0)$. This is the eigenspace corresponding to the eigenvalue of lesser magnitude (1 is less than 10) and we call it the weak eigenspace. The other eigenspace (the line through the origin with direction $(0,1)$) we call the strong eigenspace.