Math 6410 • ODE

Consider the \(T\)-periodic non-autonomous linear differential equation in \(\R^n\)
\begin{equation}
  \frac{dx}{dt}=A(t)x
  \tag{2}
\end{equation}
where \(A(t)\) is a real, smooth \(n\times n\) matrix function such that
\(A(t+T)=A(T)\) for all \(t\). Let \(\Phi(t)\) be the fundamental matrix with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] Show that there is a \(T\)-periodic function \(P(t)\) and a matrix \(B\) so that
\[
  \Phi(t)=P(t)e^{tB}.
\]
\item[(b)] Suppose the eigenvalues \(\omega\) of \(\Phi(T)\) satisfy \(\lvert\omega\rvert\le1\)
and that all \(\omega\) with \(\lvert\omega\rvert=1\) possess a full set of eigenvactors.
Show that then all solutions of (2) are bounded.
\end{enumerate}