Math 6410 • ODE

\begin{enumerate}
\item[(a)] Let \(A(t)\) be a continuous, non-constant, real matrix-valued, \(T>0\)
periodic function \(A(t+T)=A(t)\). For the system
\[
  \dot{x}=A(t)x
  \tag{2}
\]
define: \textit{Monodromy Matrix}, \textit{Floquet Multipliers} and
\textit{Floquet Exponents}.
\item[(b)] If \(e^{T\gamma}=\rho\) is a Floquet multiplier, then there is a solution
of (2) of the form
\[
  x(t)=e^{\gamma t}p(t)
\]
where \(p(t)=\pm p(t+T)\).
\item[(c)] Let \(a,b\in\R\). Suppose that a Floquet multiplier is \(\mu_1=-1\). What can
you say about the boundedness of all solutions on \([0,\infty)\)?
\[
  \begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}
  =
  \begin{pmatrix}
    -1 & a\\
    b & \sin(\pi t)
  \end{pmatrix}
  \begin{pmatrix}x\\ y\end{pmatrix}
\]
\end{enumerate}