Math 6410 • ODE

Let \(f(x)\in C^1(\R^n,\R^n)\) have a non-constant, \(T>0\) periodic trajectory
\(\gamma(t)\) satisfying \(\dot{\gamma}(t)=f(\gamma(t))\).
\begin{enumerate}
\item[(a)] Define: the periodic solution \(\gamma(t)\) is orbitally stable.
\item[(b)] Define: the Poincar\'e Map for the orbit \(\gamma\).
\item[(c)] Find a periodic solution \(\gamma(t)\) to
\[
  \begin{cases}
    \dot{x}=x-y-x(x^2+y^2),\\
    \dot{y}=x+y-y(x^2+y^2).
  \end{cases}
\]
Determine the orbital stability of \(\gamma(t)\) by computing the derivative of its
Poincar\'e Map.
\end{enumerate}