Math 6410 • ODE

Consider the system

\[
\begin{cases}
\dot{x}=x-y-x^3,\\
\dot{y}=x+y-y^3.
\end{cases}
\]

\begin{enumerate}

\item
Plot the sets \(\{\dot{x}=0\}\) and \(\{\dot{y}=0\}\) in the \((x,y)\)-plane, find the fixed
points of the system and determine their stability.


\item
Show that the annular region \(A=\{1<|(x,y)|<\sqrt{2}\}\) is forward
invariant.


\item
State the Poincar\'e-Bendixson theorem.


\item
Show that the system has at least one non-constant periodic orbit.

\end{enumerate}