Math 6410 • ODE

Consider the scalar differential equation
\[
  \ddot{x}+x=-\epsilon f(x,\dot{x})
\]
with \(\lvert\epsilon\rvert\ll1\). Let \(y=\dot{x}\).
\begin{enumerate}
\item[(a)] Show that if \(E(x,y)=(x^2+y^2)/2\) then
\[
  \dot{E}=-\epsilon f(x,y)y.
\]
Hence show that an approximate periodic solution of the form \(x=A\cos t+O(\epsilon)\)
exists if
\[
  \int_0^{2\pi} f(A\cos t,-A\sin t)\sin t\dd t=0.
\]

\item[(b)] Let \(E_n=E(x(2\pi n),y(2\pi n))\) with
\(x(t)=A_n\cos t+O(\epsilon)\) for \(2\pi n\le t<2\pi(n+1)\). Show that to lowest order
\(E_n\) satisfies a difference equation of the form
\[
  E_{n+1}=E_n+\epsilon F(E_n)
\]
and write down \(F(E_n)\) explicitly as an integral. Hence deduce that a periodic orbit
with approximate amplitude \(A^*=\sqrt{2E^*}\) exists if \(F(E^*)=0\) and this orbit is
stable if
\[
  \epsilon\frac{dF}{dE}(E^*)<0.
\]

\item[(c)] Using the above result, find the approximate amplitude of the periodic orbit
of the van der Pol equation
\[
  \ddot{x}+x+\epsilon(x^2-1)\dot{x}=0
\]
and verify that it is stable.
\end{enumerate}