Math 6410 • ODE

\begin{enumerate}
\item[(a)] Use the Poincar\'e--Bendixson theorem to establish the existence of a limit
cycle for the system
\[
\begin{aligned}
  \dot{x} &= y+\frac{x}{4}\left(1-2(x^2+y^2)\right),\\
  \dot{y} &= -x+\frac{y}{2}\left(1-(x^2+y^2)\right).
\end{aligned}
\]

\item[(b)] Consider the system
\[
\begin{aligned}
  \dot{x} &= x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}},\\
  \dot{y} &= x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}}.
\end{aligned}
\]
Show that the above pair of equations can be rewritten in polar coordinates as
\[
  \dot{r}=r(1-r^2),\quad \dot{\theta}=2\sin^2(\theta/2)
\]
and sketch the phase portrait. Determine whether or not the fixed point \((1,0)\) is
Liapunov stable.
\end{enumerate}