ODELS7

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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\,


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