ODELS6

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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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