Consider the second-order initial value problem, defined for a function $u:R\to R^n$:
$$\ddot{u} = -(I + A)u$$ $$\dot{u}(t=0) = \dot{u}_0$$ $$u(t=0) = u_0$$
In the case where $A=0$, this reduces to each component of $u$ being an independent sinusoid. Does this problem have a name for the case when $A \neq 0$? If so, are there known conditions that $A$ must satisfy for $u$ to not either diverge or converge to $0$ as $t\to\infty$?