Math 6410 • ODE

\textbf{Floquet Theory.}
Consider the system
\[
  \dot{x}(t)=A(t)x(t),
\]
where \(A:\R\to\R^{n\times n}\) is continuous and \(T\)-periodic, i.e.,
\(A(t+T)=A(t)\) for all \(t\in\R\). Let \(\Phi(t)\) be the fundamental matrix solution for
this system. Assume that there exists a complex-valued matrix \(B\) such that
\(e^{BT}=\Phi(T)\).
\begin{enumerate}
\item[(a)] Explain in detail how solutions \(x(t)\) to the periodic system above are
related to solutions \(y(t)\) of the constant-coefficient problem \(\dot{y}=By\), and prove
your assertions.
\item[(b)] Explain what the eigenvalues of \(A(t)\) or \(B\) have to do with stability of
the solutions \(x(t)\).
\end{enumerate}