Math 6410 • ODE

\textbf{Periodic Orbits.}
Consider the system
\[
\begin{aligned}
  \dot{x} &= -y+x(1-x^2-y^2),\\
  \dot{y} &= x+y(1-x^2-y^2),\\
  \dot{z} &= z.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Find the invariant manifolds for this system.
\item[(b)] Find the unique periodic orbit \(\Gamma\) for this system.
\item[(c)] Find the global stable manifold \(W^s(\Gamma)\) and global unstable manifold
\(W^u(\Gamma)\) with respect to the periodic orbit \(\Gamma\).
\end{enumerate}