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\begin{document}

\title{Differential Equations}
\author{Summer Semester 2024, Exercise 4}
\date{Due Wednesday, May 15, 2024}
\maketitle

\begin{enumerate}

\item Find all equilibrium points, determine
their stability properties, and try to sketch the phase portraits of
the following differential equations:
\begin{enumerate}
\item $\begin{aligned}[t]
         \dot x_1 & = x_2 \, (1+x_1-x_2^2)  \\
         \dot x_2 & = x_1 \, (1+x_2-x_1^2)
       \end{aligned}$
%%% This is Verhulst, Exercise 3-1. This example is nasty and should
%%% be used as a more advanced exercise.  
\item $\begin{aligned}[t]
         \dot x_1 & = 2 x_1 - x_1^2 - x_1 x_2  \\
         \dot x_2 & = -x_2 + x_1 x_2
       \end{aligned}$
\item $\ddot y + \dot y + y^3 = 0$
\end{enumerate}

\item Consider the system
\begin{align*}
  \dot x_1 & = x_2 - x_1^3 + \mu \, x_1 \,, \\
  \dot x_2 & = - x_1 \,.
\end{align*}
%%% This is the van der Pol equation, albeit not in canonical scaling
%%% Requires a rather careful and non-obvious analysis to prove
%%% existence of critical points.  Should be used as more advanced
%%% exercise.
\begin{enumerate}
\item For which values of the parameter $\mu$ does a periodic solution
exist?
\item Describe what happens as $\mu \searrow 0$.
\end{enumerate}


\end{enumerate}
\end{document}
