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  {dx_{2}}{dt}}=\left(1+\cos t-{\frac  {\sin t}{2+\cos t}}\right)x_{2}\,

Seperate variables.

{\frac  {dx_{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_{2}e^{{t+\sin t}}(2+\cos t)\,

Now use this equation in the first ODE.

x_{1}'=-x_{1}+c_{2}e^{{t+\sin t}}(2+\cos t)\,

x_{1}'+x_{1}=c_{2}e^{{t+\sin t}}(2+\cos t)\,

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

\left[e^{t}x_{1}\right]'=c_{2}e^{{2t+\sin t}}(2+\cos t)\,

Integrate.

e^{t}x_{1}=c_{2}e^{{2t+\sin t}}+c_{3}\,

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

x_{1}=c_{2}e^{{t+\sin t}}+c_{3}e^{{-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\,



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