Stability of linear differential equations with periodic coefficients

Question

I have the following system:

$$\begin{bmatrix}x'(t)\\y'(t)\end{bmatrix}=\begin{bmatrix}0&1\\-(2-\alpha\sin(t))&-(2-\alpha\cos(t))\end{bmatrix}\begin{bmatrix}x(t)\\y(t)\end{bmatrix}$$

After some simulations, I found that the solution is stable for approximately $\alpha<3.162$. However, I would like to prove it.

In this document (page 57), it says:

https://personal.math.ubc.ca/~ward/teaching/m605/every2_floquet1.pdf

However, I'm afraid this is not enough to prove stability (or I'm doing something wrong) since I get $0>-4\pi$, independently of $\alpha$.

I found another source here (page 2):

https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-16/issue-3/Stability-of-linear-differential-equations-with-periodic-coefficients-in-Hilbert/pjm/1102994832.pdf

But I have no idea how to calculate that "Gateau differential".