# ODELS5

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 $\mu _{{1,2}}=-1,1\,$