Given the ODE system

$$ \left\{ \begin{array}{l} \dot x = -2x - z \cos t, \\ \dot y = x \sin t - y, \\ \dot z = -4z + \sin^2 t. \end{array} \right. $$ I am asked to:

1) Examine stability of this system

2) Determine if the system has $2\pi$-periodic solutions.

Since we usually refer to monodromy operator in cases like this, I noticed that, given a homogeneous system $$ \left\{ \begin{array}{l} \dot x = -2x - z \cos t, \\ \dot y = x \sin t - y, \\ \dot z = -4z. \end{array} \right. $$ $z$ is an eigenvector with an eigenvalue $-4$, hence one of the monodromy eigenvalues is $e^{-4\pi}$. Then, since we can explicitly calculate $z$ from the 3rd equation, substituting it into 1st equation and considering the homogeneous part $$ \left\{ \begin{array}{l} \dot x = -2x, \\ \dot y = x \sin t - y. \end{array} \right. $$ we observe that $x$ is an eigenvector with an eigenvalue $-2$, so $e^{-2 \pi} $ is the second monodromy eigenvalue. Replacing $x$ in the 2nd equation with its explicit expression, we get the third monodromy eigenvalue: $e^{-\pi}$. Since all the monodromy eigenvalues are less than 1 (by absolute value), system is asymptotically stable.

That is my solution on the first problem. Is it correct?

And how can I solve the second problem?


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