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\begin{document}

\title{Differential Equations}
\author{Summer Semester 2024, Exercise 5}
\date{Due Wednesday, May 29, 2024}
\maketitle

\begin{enumerate}

\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 cannot be a periodic orbit unless $b>0$.

\emph{Hint:} Use Bendixon's criterion.

\item 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 $A$.

\emph{Hint:} Poincar\'e--Bendixon theorem.

\item (From Verhulst, Exercise 4.8.)  Consider the system
\begin{gather*}
  \dot x = \frac{\partial E}{\partial y} 
         + \lambda \, E \, \frac{\partial E}{\partial x} \,, \\
  \dot y = -\frac{\partial E}{\partial x} 
         + \lambda \, E \, \frac{\partial E}{\partial y} 
\intertext{with $\lambda \in \R$ and}
  E(x,y) = y^2 - 2 \, x^2 + x^4 \,. 
\end{gather*}
\begin{enumerate}
\item Put $\lambda = 0$.  Determine the critical points and their
character by linear analysis.  What happens in the nonlinear system?
Sketch the phase plane.

\item What happens to the critical points when $\lambda \neq 0$?

\item Choose $\lambda < 0$.  Determine the $\omega$-limit sets of the
orbits starting at $(\frac12,0)$, $(-\frac12,0)$, and $(1,2)$.
\end{enumerate}


\end{enumerate}
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