Math 6410 • ODE

Suppose \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\) and
periodic of period \(T>0\). Consider the linear differential equation in \(\R^n\)
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=A(t)x,\\
    x(0)=x_0.
  \end{cases}
  \tag{1}
\end{equation}
Let \(\Phi(t)\) be the fundamental matrix solution with \(\Phi(0)=I\).
\begin{enumerate}
\item[(a)] 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[(b)] Prove that the zero solution is unstable for the system
\(\dot{x}=A(t)x\), where
\[
  A(t)=
  \begin{pmatrix}
    1 & 1\\
    0 & \dot{h}(t)/h(t)
  \end{pmatrix},
\]
and \(h(t)=2+\sin t-\cos t\).
\item[(c)] Suppose that the autonomous nonlinear equation \(\dot{x}=f(x)\) exhibits
a limit cycle. Explain how Floquet theory can be used to determine the linear
stability of the limit cycle.
\end{enumerate}