Math 6410 • ODE

\begin{enumerate}
\item[(a)] Consider the van der Pol oscillator
\[
  \ddot{x}+\varepsilon\dot{x}(x^2-1)+x=0,\quad 0<\varepsilon\ll 1
\]
and choose initial conditions \(x(0)=1\), \(\dot{x}=0\). Using the method of multiple scales
show that this has an asymptotic series solution of the form
\[
  x(t)\sim R(\varepsilon t)\cos(t+\theta(\varepsilon t))+O(\varepsilon)
\]
with
\[
  \theta_\tau=0,\quad R_\tau=\frac{1}{8}R(4-R^2)
\]
where \(\tau=\varepsilon t\), \(\theta(0)=0\) and \(R(0)=1\). Hence, establish that there
exits a stable periodic orbit.

\item[(b)] Consider the Duffing equation
\[
  \ddot{x}+x=-\varepsilon x^3,\quad 0<\varepsilon\ll 1.
\]
Using the method of multiple scales show that the leading order solution is of the form
\[
  x(t)\sim R_0\cos\left(t+t_0+\frac{3}{8}R_0^2\varepsilon t\right)+O(\varepsilon).
\]
\end{enumerate}