ODELS5

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

Start with the second equation since it does not involve the first.

$\frac{d x_2}{dt} = \left(1+\cos t - \frac{\sin t}{2+\cos t}\right) x_2\,$

Seperate variables.

$\frac{d x_2}{x_2} = \left(1+\cos t - \frac{\sin t}{2+\cos t}\right) dt\,$

Integrate.

$\ln x_2 = t+\sin t + \ln(2+\cos t)+c_1\,$

Exponentiate.

$x_2 = c_2e^{t+\sin t}(2+\cos t)\,$

Now use this equation in the first ODE.

$x_1'=-x_1+c_2e^{t+\sin t}(2+\cos t)\,$

$x_1'+x_1=c_2e^{t+\sin t}(2+\cos t)\,$

Use the integrating factor $\rho = e^{\int dt} = e^t\,$

$\left[ e^t x_1 \right]' = c_2e^{2t+\sin t}(2+\cos t)\,$

Integrate.

$e^t x_1 = c_2e^{2t+\sin t}+c_3\,$

Multiply by $e^{-t}\,$ (which is okay because it's always positive).

$x_1 = c_2e^{t+\sin t}+c_3e^{-t}\,$

A fundamental matrix is $X(t)=\begin{bmatrix} \frac{\partial x_1}{\partial c_2} & \frac{\partial x_1}{\partial c_3} \\ \frac{\partial x_2}{\partial c_2} & \frac{\partial x_2}{\partial c_3}\end{bmatrix} = \begin{bmatrix} e^{t+\sin t} & e^{-t} \\ e^{t+\sin t}(2+\cos t)&0\end{bmatrix}\,$

$X^{-1}(0) = \begin{bmatrix} 0 & 1/3 \\ 1 & -1/3 \end{bmatrix} \,$

$X(2\pi) = \begin{bmatrix} e^{2\pi} & e^{-2\pi} \\ 3e^{2\pi}&0 \end{bmatrix}\,$

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

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

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

Main Page : Ordinary Differential Equations : Linear Systems

ODELS5

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