Math 6410 • ODE

Consider the \(T\)-periodic non-autonomous linear differential equation
\[
  \dot{x}=A(t)x,\quad x\in\R^n,\quad A(t)=A(t+T).
\]
Let \(\Phi(t)\) be a fundamental matrix with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] Show that there exists at least one nontrivial solution \(x=\chi(t)\) 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)] Suppose that the autonomous nonlinear equation \(\dot{x}=f(x)\) exhibits a
limit cycle. By linearizing about this solution, explain how Floquet theory can be used
to determine the linear stability of the limit cycle.
\end{enumerate}