Math 6410 • ODE

Consider the nonlinear equation
\[
  \ddot{x}+\varepsilon\left(\frac{1}{3}\bigl(\dot{x}\bigr)^3-\dot{x}\right)+x=0,
  \quad 0<\varepsilon\ll 1
\]
and choose initial conditions \(x(0)=a\), \(\dot{x}(0)=0\).
\begin{enumerate}
\item[(a)] Using the method of multiple scales show that this has an asymptotic series
solution of the form
\[
  x(t)\sim 2R(\varepsilon t)\cos(t+\theta(\varepsilon t))+O(\varepsilon)
\]
with
\[
  \theta_\tau=0,\quad R_\tau=\frac{1}{2}R(1-R^2)
\]
where \(\tau=\varepsilon t\).

\item[(b)] Derive the solution
\[
  R(\tau)=\left(1+a_0e^{-\varepsilon t}\right)^{-1/2}
\]
and determine \(a_0\) from the initial conditions. Hence, establish that there exits a
stable periodic orbit.
\end{enumerate}