Math 6410 • ODE

\begin{enumerate}
\item[(i)] Suppose \(A\) is an \(n\times n\) matrix whose elements \(a_{ij}(t)\) are each of
period \(T\). You may assume existence and uniqueness of a fundamental matrix
\(\Phi(t)\) for the equation
\[
\dot{x}(t)=A(t)x(t).
\]
Show that \(\Phi(t+T)\) is also a fundamental matrix for the equation, hence
\[
\Phi(t+T)=\Phi(t)\cdot C
\]
for some non-singular matrix \(C\).

\item[(ii)] You may assume that a non-singular matrix \(C\) has a logarithm, i.e.,
\(C=e^B\) for some \(n\times n\) matrix \(B\).

Show that there exists a non-singular matrix \(P(t)\) of period \(T\) such that
\[
\Phi(t)=P(t)e^{Bt}
\qquad\text{Floquet's Theorem.}
\]

\textbf{Hint:} Solve for \(P(t)\) and check.

\item[(iii)] Make the change of variables \(y=P(t)u(t)\) and change the periodic system
\(y'=A(t)y\) into the system
\[
u'=Bu.
\]
\end{enumerate}