Math 6410 • ODE

Consider the \(T\)-periodic non-autonomous linear differential
equation

\[
\dot{x}=A(t)x,\quad x\in\mathbb{R}^n,\quad A(t)=A(t+T)
\]

Let \(\Phi(t)\) be a fundamental matrix with \(\Phi(0)=I\).
\begin{enumerate}

\item
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
Suppose that \(\Phi(T)\) has \(n\) distinct eigenvalues \(\mu_i\), \(i=1,\ldots,n\). Show
that there are \(n\) 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
Consider the equation \(\dot{x}=f(t)A_0x\), \(x\in\mathbb{R}^2\), with \(f(t)\) a scalar
\(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigen-
values. Determine the corresponding Floquet multipliers.

\end{enumerate}