differential equations solve for exponents Consider $x' = A(t)x$, $x \in \mathbb{R}^n$ where $A$ is $2\pi$-periodic. 
$$A(t) = \begin{bmatrix} 1+\sin(t)&0&0\\ 0&3&4\\ 0&1&3\end{bmatrix}$$
The question is to find the Floquet exponents and it also asked to find Lyapunov exponents. I am kind of new to ordinary differential equations, and it would be very nice to show step by steps to solve this problem to firmly understand the concept.
 A: The original system consists of 2 parts:
$$\tag{1}
\dot x=(1+\sin t)x
$$
and
$$\tag{2}
\frac{d}{dt}\left(\begin{array}{c}y\\z\end{array}\right)=
B\left(\begin{array}{c}y\\z\end{array}\right),\qquad B=
\left(\begin{array}{cc}
3&4\\1&3
\end{array}\right).
$$
We can solve them separately, obtaining
$$
x(t)=C_1e^{t-\cos t}
$$
$$
\left(\begin{array}{c}y\\z\end{array}\right)=
e^{Bt}\left(\begin{array}{c}C_2\\C_3\end{array}\right)=
\left(\begin{array}{cc}
 e^{5t}/2 + e^t/2&     e^{5t} - e^t\\
 e^{5t}/4 - e^t/4& e^{5t}/2 + e^t/2
\end{array}\right)\left(\begin{array}{c}C_2\\C_3\end{array}\right);
$$
or
$$
\left(\begin{array}{c}x\\y\\z\end{array}\right)=
\left(\begin{array}{ccc}
e^{t-\cos t}&0&0\\
0& e^{5t}/2 + e^t/2&     e^{5t} - e^t\\
0& e^{5t}/4 - e^t/4& e^{5t}/2 + e^t/2
\end{array}\right)\left(\begin{array}{c}C_1\\C_2\\C_3\end{array}\right)=X(t)
\left(\begin{array}{c}C_1\\C_2\\C_3\end{array}\right).
$$
The fundamental matrix $X(t)=\left(\begin{array}{c|c}
e^{t-\cos t}&0\\ \hline
0&e^{Bt}
\end{array}\right)$ can be written in the Floquet's canonical form 
$$
X(t)=\Phi(t)e^{\Lambda t},
$$
where
$$
\Phi(t)=
\left(\begin{array}{ccc}
e^{-\cos t}&0&0\\
0& 1&     0\\
0& 0& 1
\end{array}\right)
$$
is a $2\pi$ periodic matrix,
$$
e^{\Lambda t}=
\left(\begin{array}{c|c}
e^t&0\\ \hline
0&e^{Bt}
\end{array}\right)=
e^{
\left(\begin{array}{c|c}
t&0\\ \hline
0&Bt
\end{array}\right)}=
e^{
\left(\begin{array}{c|c}
1&0\\ \hline
0&B
\end{array}\right)t},
$$
hence,
$$
\Lambda=
\left(\begin{array}{ccc}
1&0&0\\
0& 3&  4\\
0& 1& 3
\end{array}\right).
$$
$\Lambda$ has the eigenvalues  $1,1,5$, thus,
the Floquet multipliers of the system are the eigenvalues of 
$e^{2\pi\Lambda}$, i.e. $e^{2\pi}$, $e^{2\pi}$, $e^{10\pi}$;
the Floquet exponents are $1,1,5$; The Lyapunov exponents are the real parts of the Floquet exponents, so they are equal to $1,1,5$.
