Math 6410 • ODE

Consider the \(T\)-periodic non-autonomous linear differential equation
\[
  \dot{x}=A(t)x,\qquad x\in\R^n,\qquad A(t)=A(t+T)
\]
Let \(\Phi(t)\) be a fundamental matrix with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] Show that there exists at least one nontrivial solution \(\chi(t)\) such that
\[
  \chi(t+T)=\mu\chi(t)
\]
where \(\mu\) is an eigenvalue of \(\Phi(T)\).
\item[(b)] 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[(c)] By determining the time derivative of the Wronskian
\(W(t)=\det[\Phi(t)]\), prove that the product of the Floquet multipliers \(\mu_i\)
satisfies
\[
  \prod_{i=1}^n\mu_i
  =\exp\left(\int_0^T\operatorname{tr}[A(s)]\dd s\right)
\]
\item[(d)] Using part (c) show that \(\mu_1\mu_2=1\) in the case of Mathieu's equation
for a parametric oscillator:
\[
  \ddot{x}+(\alpha+\beta\cos t)x=0.
\]
\end{enumerate}