Math 6410 • ODE

Consider the nonlinear equation
\[
  \ddot{x}+\epsilon\dot{x}(x^2-1)+x=0,\qquad 0<\epsilon\ll1
\]
and choose initial conditions \(x(0)=1\), \(\dot{x}(0)=0\).
\begin{enumerate}
\item[(a)] Using the method of multiple scales show that this has an asymptotic series
solution of the form
\[
  x(t)\sim R(\epsilon t)\cos(t+\theta(\epsilon t))+\mathcal{O}(\epsilon)
\]
with
\[
  \theta_\tau=0,\qquad R_\tau=\frac{1}{8}R(4-R^2)
\]
where \(\tau=\epsilon t\).
\item[(b)] Derive the solution
\[
  R(\tau)=\frac{2}{(1+a_0e^{-\epsilon t})^{1/2}}
\]
and determine \(a_0\) from the initial conditions. Hence, establish that there exits
a stable periodic orbit. [HINT: perform the change of variable \(y=R^2\).]
\end{enumerate}