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
\[
  \dot{x}=A(t)x
  \tag{4}
\]
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 (4) of the form
\[
  x(t)=e^{\gamma t}p(t)
\]
where \(p(t)=\pm p(t+T)\).
\item[(c)] Prove that this equation does not have a fundamental set of bounded
solutions.
\[
  \ddot{x}-(\cos^2t)\dot{u}+(\sin^2t)u=0
\]
\end{enumerate}