# 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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## 1 Answer

It's often helpful to split stable manifolds into strong and weak parts, particularly for models in physiology and chemistry for which different mechanisms act on different time-scales.

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.

More generally for nonlinear systems with a similar eigenvalue separation, solutions rapidly approach a curve (or curved surface) instead of a straight line, and we call the curve (or surface) a weak stable manifold, or a slow manifold. Similarly, corresponding to the eigenvalue of greater magnitude we have a strong stable manifold. The strong stable manifold is tangent to the strong eigenspace.

The theory for strong and weak manifolds is not as clear-cut as that for stable and unstable manifolds. Often they are not unique. Also, if your eigenvalues are -1, -10 and -100, you may wish to identify three levels of manifold strength.

Of the various dynamical systems textbooks out there, you might like to look in the book by Perko and the book by Chicone.

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Thank you for the answer! Not only do you provide a definition, but also the reference, which is very nice of you. I am strongly surprised by the definition, but in practical applications I trust it must make sense. –  Feanor Oct 17 '12 at 20:54