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 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
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
eigenvalues. Determine the corresponding Floquet multipliers.


\item
Suppose that the autonomous nonlinear equation \(\dot{x}=f(x)\),
\(x\in\mathbb{R}^d\), 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}