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Suppose I have a nonlinear system of ODEs and a connected manifold $M$ of fixed points. I want to show that $M$ is asymptotically stable. Now, if I pick a point on $M$ and linearize the system around this point, I will obviously get zero eigenvalues corresponding to $M$'s tangent vectors (by assumption we have $\dot x = 0$ everywhere on $M$, so to first order, $\dot x$ is constant when moving in a direction tangent to $M$). Thus, stability cannot be established by the method for hyperbolic fixed points.

However, suppose that at all points on $M$, all other eigenvalues (i.e., all eigenvalues for eigenvectors not in $TM$) have real part $< 0$. It seems intuitively clear that then, distance to $M$ should be a (decreasing) local Lyapunov function: at each $\hat x\in M$, if we move a small enough distance in a direction orthogonal to $M$, the negative eigenvalues seem to tell us that the component of $\dot x$ orthogonal to $TM$ will bring us closer to $M$, while the zero eigenvalues just tell us that we can't say anything about the component of $\dot x$ along M.

Is this in fact correct? And what would be some keywords for looking this up on Google or in a book?

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