# ODELS6

Find a fundamental matrix and characteristic multipliers and exponents for the system $\begin{cases}x_1'=(1+2\cos 2t) x_1 + (1-2\sin 2t)x_2 \\ x_2' = -(1+2\sin 2t)x_1 + (1-2\cos 2t)x_2\end{cases}\,$

The fundamental matrix is $X(t)=\begin{bmatrix} e^{3t}\cos t & e^{-t}\sin t \\ -e^{3t}\sin t & e^{-t}\cos t\end{bmatrix}\,$

$X^{-1}(0) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \,$

$X(\pi) = \begin{bmatrix} -e^{3\pi} & 0 \\ 0 & -e^{-\pi} \end{bmatrix}\,$

The monodromy matrix is $B = X^{-1}(0)X(\pi) = \begin{bmatrix} e^{3\pi} & 0 \\ 0 & e^{-\pi} \end{bmatrix}\,$

The characteristic multipliers are $\rho_{1,2}=e^{3\pi},e^{-\pi}\,$

The characteristic exponents are $\mu_{1,2} = 3,-1\,$

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