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)] Consider the equation \(\dot{x}=f(t)A_0x\), \(x\in\R^2\), with \(f(t)\) a scalar
\(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigenvalues.
Determine the corresponding Floquet multipliers.
\end{enumerate}