Lyapunov Stability Theorem states, loosely speaking, that for $x' = Ax + G(x,t)$ where $G(x, t) = o(\|x\|)$, if all eigenvalues of $A$ have negative real parts, then solution $x_0=0$ is asymptotically stable.
It seems obvious to me that if at least one eigenvalue has a positive real part, then $x_0 = 0$ is not asymptotically stable (since, again, we can simply ignore $G(x,t)$ when $x \to 0$). However, I wasn't able to find a reference. Is this statement true? Does it have a name?