In wikipedia https://en.wikipedia.org/wiki/Mathieu_function#Floquet_solution
I want to know how the Floquet solution is plotted.
One way I am thinking is to write Floquet solution in terms of the elemental solutions: Mathieu sin and cos, so that I can call these two functions built in software like Mathmatica. But I don't How to write Floquet solution in terms of Mathieu sin and cos.
I'm not quite sure what's going on in that plot. The basics of the Floquet solution are this. You start off with the first order differential equation system for the fundamental matrix $$ \phi'(t) = A(t) \phi(t), \ \phi(0) = I $$ where in this case $$A(t) = \pmatrix{ 0 & 1\cr -a + 2 q \cos(2t) & 0\cr} $$ which is periodic with period $\pi$. In this case that fundamental matrix is $$ \phi(t) = \pmatrix{ C(a,q,t) & S(a,q,t)\cr C'(a,q,t) & S'(a,q,t)\cr} $$ where $C$ and $S$ are the Mathieu cosine and sine functions, $C'$ and $S'$ their derivatives with respect to $t$. Now we let $B$ be a matrix such that $e^{\pi B} = \phi(\pi)$. In this case we can take $$ B = \pmatrix{ i & .100721974647390\cr 0.0983648541461455 & i\cr}$$ Floquet says that the matrix function $P(t) = \phi(t) e^{-tB}$ is periodic. If $v$ is an eigenvctor of $B$ for eigenvalue $\lambda$, then $w(t) = \phi(t) v$ is a solution of $w'(t) = A(t) w(t)$ with $w(\pi) = e^{\pi \lambda} v = e^{\pi \lambda} w(0)$. In our case $B$ has real eigenvectors
$$ \eqalign{\pmatrix{0.711280416113229\cr 0.702908365047530} &\ \text{for eigenvalue}\ 0.0995364372755138 + i\cr \pmatrix{0.711280416113229\cr -0.702908365047530} &\ \text{for eigenvalue}\ -0.0995364372755138 + i\cr}$$
The plots of the first entry of $w(t)$ (corresponding to solutions of the Mathieu equation) for these eigenvectors look like this (the first one in red, the second in green):
The first describes a solution that, after one period ($t=\pi$) has changed sign (because of the imaginary part $i$ in the eigenvalue) and grown in amplitude by a factor $e^{0.0995364372755138 \pi} = 1.36711535556321$. The second also changes sign, but has decreased in amplitude: in fact it is the time-reversal of the first.