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Given a system of ODEs,

$\mathbf{x}' = A\mathbf{x}$,

one way to determine the stability of an equilibrium point is to look at the eigenvalues of the Jacobian matrix. However, there are cases in which this test won't immediately give conclusive information (such as when the real part of one eigenvalue is zero and the real parts of the others are negative, or when there is a mix of positive and negative real parts). If this situation arises in a specific example, I've always just used a somewhat ad hoc approach to determine stability.

My question is the following: Is there a general technique for approaching such a system when the "eigenvalue test" fails, or does one usually just have to use an example-specific approach?

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    I'm not sure I understand how looking at eigenvalues is inconclusive in those cases you wrote. If any of the eigenvalues have positive real part, then some trajectory will go out to infinity, so it is unstable. It is stable in the other case you state. Is the question determining the difference between asymptotically stable and stable in that case?2012-10-25
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    In the nonlinear case $\textbf{x}' = \textbf{f}(\textbf{x})$, centers are inconclusive (i.e. no [Hartman-Grobman](http://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman_theorem)).2012-10-26

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