I have to find conditions under which the solutions of x'=Ax are bounded for $t\rightarrow \infty$ and $t\rightarrow \pm \infty$. We proved in our course that $\lim_{t \rightarrow \ \infty} x(t)=0$ and $\lim_{t \rightarrow - \infty} |x(t)|=\infty$, if all the eigenvalues of $A$ have negative real part (and for the last limit $x$ is not the zero solution); if $A$ has an eigenvalue greater that zero, then $\lim_{t \rightarrow \infty} |x(t)|=\infty$.
So I think that the conditions are: $x$ is bounded for $t\rightarrow \infty$ , if the real part of all the eigenvalues are negative (trivial, since it follows from my course) and bounded for $t\rightarrow \pm \infty$ if the real parts of all the eigenvalues vanishes. Do you think this is sufficient - or that my professor could want some other additional (finer) conditions ?
Side question: Since we didn't specify what $|x(t)|$ means, I took it means the norm of $x(t)$ and not the vector of the absolute values of the components of $x(t)$, because in the latter case $\lim_{t \rightarrow \infty} |x(t)|=\infty$ doesn't makes sense to me, because only a sequence of numbers can have the limit $\infty$. Am I right that it means the norm ?