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 \(\R^n\)
\[
  \begin{cases}
    \dfrac{dx}{dt}=A(t)x,\\
    x(0)=x_0.
  \end{cases}
  \tag{2}
\]
Let \(\Phi(t)\) be the fundamental solution matrix with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] Suppose that \(\Phi(T)\) has \(n\) distinct eigenvalues \(\mu_i\),
\(i=1,\ldots,n\). Show that there are then \(n\) linearly independent solutions of
the form
\[
  x_i=p_i(t)e^{\rho_i t}
\]
where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho_i\) related to \(\mu_i\)?
\item[(b)] What is the fundamental solution matrix and what are the Floquet
exponents for the system
\[
  A(t)=
  \begin{pmatrix}
    a & 1\\
    0 & \dfrac{\dot{p}(t)}{p(t)}
  \end{pmatrix},
  \tag{3}
\]
where \(p(t)\) is a \(T\)-periodic function?
\end{enumerate}