ODELS7

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

The fundamental matrix is $X(t)=e^t\begin{bmatrix} 2+\sin t & 0 \\ 4t-2\cos t & 1 \end{bmatrix}\,$

$X^{-1}(0) = \begin{bmatrix} 1 & 0 \\ 2 & 2 \end{bmatrix} \frac{1}{2} = \begin{bmatrix} \frac{1}{2} & 0 \\ 1 & 1 \end{bmatrix}\,$

$X(2\pi) = e^{2\pi}\begin{bmatrix} 2 & 0 \\ 8\pi-2 & 1 \end{bmatrix}\,$

The monodromy matrix is $B = X^{-1}(0)X(2\pi) = e^{2\pi}\begin{bmatrix} 1 & 0 \\ 4\pi & 1 \end{bmatrix}\,$

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

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