What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$? What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$? The best I can come up with is that $A=bi\cdot M$ for $b$ a non-zero real number and $M$ a non-zero idempotent matrix, since in that case we have: $$\|e^{tA}\|_2 = \left\|\sum_{i=0}^{\infty}\frac{(tbi)^k\cdot M^k}{k!}\right\|_2= \left\|I+M\sum_{i=1}^{\infty}\frac{(tbi)^k}{k!}\right\|_2= \|I+Me^{tbi}\|_2,$$ which quite clearly oscillates. However it's certainly not clear to me that this is the only way to obtain oscillation, and if it is, I don't see how to go about proving it.

Most non-normal matrices whose eigenvalues are pure imaginary will have this property. (Normal means $AA^* = A^*A$.)

E.g. $A = P D P^{-1}$ where $P$ is an invertible $2\times 2$ matrix that is not a multiple of a unitary matrix, and $D = \begin{bmatrix} 0 & -1 \\\ 1 & 0 \end{bmatrix}$: then $$ e^{tA} = P \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\\ \sin(\theta) & \cos(\theta) \end{bmatrix} P^{-1} = \cos(\theta)I + \sin(\theta) B ,$$ with $B = P D P^{-1}$.

E.g. $A = \begin{bmatrix} 0 & \lambda-1 \\\ \lambda+1 & 0 \end{bmatrix}$ if $-1 < \lambda < 1$.