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{\LARGE {Mathematics 118A, Winter 1998}}
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\large Homework Assignment 6, due Monday, March 2
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\begin{enumerate}

\item The Lienard form of the Van der Pol equation,
\begin{align*}
  \dot x_1 & = x_2 - (\tfrac13 x_1^3 - x_1) \\
  \dot x_2 & = -x_1,
\end{align*}
is a special case of the system characterized by Theorem 4.6 in
Verhulst's book.  Re-do the proof of this Theorem for this special
case only.  You should strive for a very clear exposition of the
argument.

\item What happens if you add a fifth order term to the Van der Pol
equation?  Consider the system
\begin{align*}
  \dot x_1 & = x_2 - (\tfrac15 x_1^5 - x_1^3 + x_1) \\
  \dot x_2 & = -x_1 .
\end{align*}
Is this case still covered by Theorem 4.6?  How many periodic
solutions to you expect to find?  Use \emph{Mathematica} to plot the
periodic orbits in the phase space.  (How do you find these orbits if
you can't solve the equation explicitly?)  Try to conjecture how the
addition of higher order terms to the system will change the phase
portrait.



\end{enumerate}
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