In this question about Floquet theory the author asked about the fundamental matrix solution $X(t)$ of the following $2\pi$-periodic differential equation $${\displaystyle {\dot {x}}=A(t)x}$$ with
$$A(\text{t})\text{=}\left( \begin{array}{cc} \sin (t) & \sin ^2(t) \cos (t) \\ \sin ^2(t) \cos (t) & \cos (t) \\ \end{array} \right);$$
In the answers, the matrix (monodromy matrix) $X(2\pi) = B$ was calculated using Maple's numerical DE solver: $$ B = \pmatrix{0.736606947094663 & 0.310166738881922\cr -3.21321753950662 &0.00457171990219575\cr}$$ Its eigenvalues are $e^{\pm ic}$ where $c/\pi \approx 0.3791557842561098$ is not rational because it does not have a small denominator: the continued fraction starts $[0; 2, 1, 1, 1, 3, 7, 2, 2, 1, 1, 10, 3]$.
How can we do such computations in Mathematica, especially $B=X(2\pi)$ and $c$?