Math 6410 • ODE

\begin{enumerate}
\item[(a)] Give definitions for the following: \textit{invariant set},
\textit{attracting set}, \textit{\(\omega\)-limit set}.
\item[(b)] Use the Poincare'-Bendixson (PB) Theorem and the fact that the planar system
\[
  \dot{x}=x-y-x^3,\qquad \dot{y}=x+y-y^3
\]
has only the one critical point at the origin to show that this system has a periodic
orbit in the annular region \(A=\{x\in\R^2\mid 1<\abs{x}<\sqrt{2}\}\).
\item[(c)] Show that the system
\[
  \dot{x}=x-rx-ry+xy,\qquad \dot{y}=y-ry+rx-x^2
\]
can be written in polar coordinates as \(\dot{r}=r(1-r)\),
\(\dot{\theta}=r(1-\cos\theta)\). Show that it has an unstable node at the origin
and a saddle node at \((1,0)\). Use this information and the PB Theorem to sketch
the phase portrait and then deduce that for all \((x,y)\ne(0,0)\), the flow
\(\phi_t(x,y)\to(1,0)\) as \(t\to\infty\) but that \((1,0)\) is not linearly stable.
\end{enumerate}