System of first order ODEs with coherent sinusoidal time varying coefficient

Question

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square matrices with constant (and real) elements. I am interested in the general properties of the unknowns at large times (after transients die away). I have already observed that while stability of the solutions is trivially simple for one degree of freedom systems (requiring ${A_0} < 0$), this criteria has no obvious extension to higher degree systems (to me at least). I would be grateful if someone could points out some references or helpful hints in studying these type of ODEs.