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\,


Main Page : Ordinary Differential Equations : Linear Systems

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