Given an ODE $x'=f(x)$ (in a open subset of $\mathbb{R}^n$) and a singularity $x_0$ (point where $f(x_0)=0$), I learned the following sufficient conditions for classifying stability:
Let $A=f'(x_0)$ be the jacobian matrix at the singularity.
i) If $A$ has a eigenvalue with positive real part, then $x_0$ is unstable;
ii) If every eigenvalue has (strictly) negative real part, then $x_0$ is assymptotically stable;
What can we say if there is at least one eigenvalue with real part 0 and the remaining have negative real part (so it doesn't fit either case)?
If we can't conclude anything, what are some examples of each possible case:
Stable, but not assymptotically;
Assymptotically stable;
Unstable;