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?