Math 6410 • ODE

\begin{enumerate}
\item[(a)] For a general dynamical system, give the definitions of an
\textit{invariant set}, \textit{attracting set} and \(\omega\)-\textit{limit sets}.
\item[(b)] For the system
\[
  \dot{x}=-f(x,y)x-g(x,y)y,\qquad
  \dot{y}=g(x,y)x-f(x,y)y,
  \tag{6}
\]
where \(g(x,y)>0\) for all \(x,y\), and \(f(x,y)=x^2+y^2+2y-1\), identify all critical
points and their stability, and show that this system has an invariant annular
region which encloses the origin, and therefore has a nontrivial periodic solution.
\end{enumerate}