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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\Re{\operatorname{Re}}

\begin{document}

\title{Differential Equations}
\author{Homework 5}
\date{Due in class Thursday, March 22, 2018}
\maketitle

\begin{enumerate}


\item Consider the system
\begin{gather*}
  \dot x = 1 + y - x^2 - y^2 \,, \\
  \dot y = 1 - x - x^2 - y^2 \,.
\end{gather*}
\begin{enumerate}
\item Determine the critical points and their character.
\item Show that the flow generated by this equation is symmetric with
respect to the line $x + y = 0$.
\item Find an explicit periodic solution $\phi(t)$

\emph{Hint}: Write the equation in polar coordinates.

\item Derive the linearization of the equation about the periodic
solution $\phi(t)$.

\item Determine both Floquet exponents of the linearized, time-periodic,
system obtained in part (d).

\item Sketch the phase portrait of the system. 

\emph{Hint}: Introduce new variables $u = x - y$ and $v = x + y$ that
exploit the symmetry and find a first integral for the system in $u$
and $v$ coordinates (introduce $w = v^2$ and solve the equation for
$\d w/\d w$ to obtain the first integral).

\item Show that $\phi(t)$ is stable, but not asymptotically stable.

\item Can you find a small perburbation to the system in the form
\begin{gather*}
  \dot x = 1 + y - x^2 - y^2 + \varepsilon \, g(x,y) \,, \\
  \dot y = 1 - x - x^2 - y^2 + \varepsilon \, h(x,y) \,,
\end{gather*}
which preserves the periodic orbit $\phi(t)$ and makes it
asymptotically stable.  Establish the asymptotic stability by
computing the perturbed Floquet exponents.
\end{enumerate}

\item (Verhulst, Exercise 8.4.) Consider the system
\begin{gather*}
  \dot x = 2 \, y \, (z-1) \,, \\
  \dot y = -x \, (z-1) \,, \\
  \dot z = x \, y \,,
\end{gather*}
\begin{enumerate}
\item Show that the trivial solution is stable.
\item Is it asymptotically stable?
\end{enumerate}

\item (Verhulst, Exercise 8.5.) 
Determine the stablity of the trivial solution of the system
\begin{gather*}
  \dot x = 2 \, x \, y + x^3 \,, \\
  \dot y = x^2 - y^5 \,.
\end{gather*}


\end{enumerate}





\end{document}
