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
\[
  \dot{x}=x-y-x^3,\qquad \dot{y}=x+y-y^3,
\]
identify all critical points and their stability, and use the Poincare-Bendixson
theorem to show that this system has a periodic orbit in the annular region
\(1<x^2+y^2<2\).
\end{enumerate}