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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? – Matt Oct 25 '12 at 20:14
In the nonlinear case $\textbf{x}' = \textbf{f}(\textbf{x})$, centers are inconclusive (i.e. no Hartman-Grobman)‌​. – Pragabhava Oct 26 '12 at 1:06

The general statement goes as follows:

  1. If all eigenvalues have negative real part then the origin is asymptotically stable
  2. If at least one of the eigenvalues has positive real part the origin is unstable
  3. If some of eigenvalues have negative real part, and some have real parts equal to zero then the origin is stable if any of the eigenvalues with zero real part has $k$ linearly independent eigenvectors assuming that multiplicity of this eigenvalue is $k$. Otherwise the origin is unstable.
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