Suppose $\dot{x}(t) = f(x(t))$ and let $x^*$ be a critical point, i.e., $f(x^*) = 0$. Let us consider the linearization of $f$ at $x^*$. Thus, we have $\dot{x}(t) = A(x^*)x(t)$. Suppose the eigenvalues of $A^*=A(x^*)$ are always non-positive (i.e., always less than or equals to $0$).
If all eigenvalues of $A^*$ are negative, we say $x^*$ is linearly stable. Suppose $A^*$ has zero eigenvalues. Then, what can we say about the linear stability of $x^*$?
Is there any name referring to such a case? It seems that the notion of marginal stability is suitable. I think it is appropriate calling such a $x^*$ as linearly marginally stable. However, I couldn't find precise references that use such terminology.
The rests are the collections of what I have searched:
Wikipedia page of linear stability says
If there exists an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation.
It seems that the notion of the center manifold is the most relevant notion, however, it simply refers to the eigenspace of $A(x^*)$ that corresponds to $\lambda = 0$, and does not itself tell the (linear) stability of $x^*$.
Any suggestions/comments/answers will be very appreciated.