Yes, it has been extensively investigated and depends upon the eigenvalues. If all eigenvalues are positive, you have an unstable point. If all eigenvalues are negative, you have a stable point. If some eigenvalues are positive and some negative, you have a kind of "saddlepoint" which you can "approach" along directions corresponding to eigenvectors with negative eigenvalues and move away from along directions corresponding to eigenvectors with positive eigenvalues.