Math 6410 • ODE

Consider the torqued pendulum with constant drag \(\beta>0\) and torque \(\mu>1\) large
enough to overcome the pull of gravity, no matter what the angle \(x\) of the pendulum
may be.
\[
  \begin{aligned}
    \dot{x}&=y\\
    \dot{y}&=\mu-\beta y-\sin x
  \end{aligned}
\]
\begin{enumerate}
\item[(a)] Show that the solution of the initial value problem with \(x(0)=0\) and
\(y(0)=y_0\) exists for all time.
\item[(b)] Show that there is a solution such that the pendulum continues to rotate
indefinitely in a \(T\)-periodic fashion. It would have
\(x^*(T+t)=x^*(t)+2\pi\) and \(y^*(T+t)=y^*(t)\).
\item[(c)] Show that the periodic solution is unique.
\item[(d)] Show that the periodic solution is orbitally stable.
\end{enumerate}