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}
  \frac{dx}{dt}=A(t)x
  \tag{1}
\end{equation}
Let \(\Phi(t)\) be the fundamental matrix solution with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] Let \(\zeta(t)\in C^1([0,\infty),\R^n)\) be a solution of (1).
Define: \(\zeta(t)\) is a \textit{Stable} solution. (This notion is also sometimes
called \textit{Liapunov Stable}.)
\item[(b)] Show that the zero solution of (1) is stable if and only if every eigenvalue
\(\omega\) of \(\Phi(T)\) satisfies either \(\lvert\omega\rvert<1\), or
\(\lvert\omega\rvert=1\) but in this case \(\omega\) appears only in matrices \(J_i\)
(from the Jordan Canonical Form for \(\Phi(T)\)) such that \(J_i\) is a \(1\times1\)
matrix.
\end{enumerate}