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\begin{document}

\title{Differential Equations}
\author{Homework 3}
\date{Due in class Tuesday, March 6, 2018}
\maketitle

\begin{enumerate}

\item Recall that the $\omega$-limitset of an orbit $\gamma(x)$
originating at a phase point $x$ is the set of all accumulation points
of the \emph{forward} orbit $\gamma^+(x)$.  I.e.,
$y \in \omega(\gamma)$ if there exists a sequence $t_n \to \infty$
such that $\phi_{t_n}(x) \to y$.

\begin{enumerate}
\item Prove that $\omega(\gamma)$ is closed and invariant.  
\item If the forward orbit is bounded, then $\omega(\gamma)$ is
compact, non-empty, and connected.
\end{enumerate}

\item Recall the Poincar\'e--Bendixon Theorem (the proof will be
finished during the next class meeting): For an autonomous
differential equation in the plane, suppose a forward orbit
$\gamma^+(x)$ is bounded and $\omega(\gamma)$ contains no critical
points.  Then $\omega(\gamma)$ is either a periodic orbit, equal to
$\gamma(x)$, or a limit cycle so that
$\omega(\gamma) = \overline{\gamma^+(x)} \ \gamma^+(x)$.

In this setting, prove the following: Suppose $\gamma_1$ and
$\gamma_2$ are two periodic orbits with $\gamma_2$ in the interior of
$\gamma_1$.  Suppose further that there are no critical points or
periodic orbits in the annular region $A$ between $\gamma_1$ and
$\gamma_2$.  Show that, for $x \in A$, $\omega(\gamma(x))$ is either
$\gamma_1$ or $\gamma_2$.  Show further that the $\omega$-limitset is
the same for all orbits in $A$.


\item Consider the system
\begin{gather*}
  \dot x_1 = x_2 \,, \\
  \dot x_2 = a \, x_1 + b \, x_2 - x_1^2 \, x_2 - x_1^3 \,.
\end{gather*}
Show that there is cannot be a periodic orbit unless $b>0$.


\item Show that the van der Pol equation
\begin{gather*}
  \dot x_1 = x_2 \,, \\
  \dot x_2 = - x_1 + \lambda \, (1-x_1^2) \, x_2  
\end{gather*}
has a non-trivial period orbit for all $\lambda \in \R$.

\end{enumerate}




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