Math 6410 • ODE

Let \(f(x)\in C^1(\mathbb{R}^n,\mathbb{R}^n)\) have a non-constant,
\(T>0\) periodic trajectory \(\gamma(t)\) satisfying

\[
\dot{\gamma}(t)=f(\gamma(t)).
\]

\begin{enumerate}

\item
[1] Define: the periodic solution \(\gamma(t)\) is orbitally stable.


\item
[2] Define: the Poincare Map for the orbit \(\gamma\).


\item
\([17]\) \(\gamma(t)=(2\cos 2t,\sin 2t)\) is a periodic solution to
\[
\begin{cases}
\dot{x}=-4y+x\left(1-\dfrac{x^2}{4}-y^2\right),\\
\dot{y}=x+y\left(1-\dfrac{x^2}{4}-y^2\right).
\end{cases}
\]
Determine the orbital stability of \(\gamma(t)\) by computing the derivative of its Poincare Map.

\end{enumerate}