I am trying to calculate whether the system $\mathbf{y}'=A(t)\mathbf{y}$ is stable, asymptotically stable, or unstable.
$ A(t)=\left(\begin{array}{cc}{-\frac{1}{4}+\frac{3}{4} \cos 2 t} & {1-\frac{3}{4} \sin 2 t} \\ {-1-\frac{3}{4} \sin 2 t} & {-\frac{1}{4}-\frac{3}{4} \cos 2 t}\end{array}\right)$
The eigenvalues of this matrix are :
$\lambda_1=\frac{1}{4}(-1+i \sqrt{7})\\ \lambda_2=\frac{1}{4}(-1-i \sqrt{7})$
This matrix is bounded so $||A(t)|| < C$ for all t and $\mathcal{R}e(\lambda_i)<0$ for all eigenvalues, so atleast it is not unstable. Also note that $|A(t)|\neq 0$ for $t\rightarrow\infty$.
So is this matrix stable or asymptotically stable, I am not sure of the difference.