Math 6410 • ODE

Suppose \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\) and
periodic of period \(T>0\). Consider the linear differential equation in \(\R^n\)
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=A(t)x,\\
    x(0)=x_0.
  \end{cases}
  \tag{1}
\end{equation}
Let \(\Phi(t)\) be the fundamental matrix solution with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] Define: \textit{Floquet Matrix}, \textit{Floquet Multiplier} and
\textit{Floquet Exponent}. How are these related to \(\Phi(t)\)? State the necessary
and sufficient conditions so that (1) has a nonzero \(T\)-periodic solution.
\item[(b)] Prove that the zero solution is unstable for the system \(\dot{x}=A(t)x\),
where
\[
  A(t)=
  \begin{pmatrix}
    1 & 1\\
    0 & \dot{h}(t)/h(t)
  \end{pmatrix},
\]
and \(h(t)=2+\sin t-\cos t\).
\end{enumerate}