Math 6410 • ODE

\begin{enumerate}
\item[(a)] Give the definitions of Poincare and Liapunov stability. Show that solutions
of the system \(\dot{x}=y\), \(\dot{y}=0\) are Poincare but not Liapunov stable.

\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}