Math 6410 • ODE

Suppose that \(f:\R^2\times\R\to\R^2\) is a smooth function and
\(\Gamma_0\subset\R^2\) is a nonconstant periodic orbit of the parameter dependent
autonomous system for the parameter value \(\epsilon=\epsilon_0\),
\begin{equation}
  \frac{dx}{dt}=f(x,\epsilon_0).
  \tag{3}
\end{equation}
\begin{enumerate}
\item[(a)] Define the following: \(\Sigma\) is a Poincar\'e section for for
\(\Gamma_0\) at the point \(x_0\in\Gamma_0\) and \(P:\Sigma\to\Sigma\) is the Poincar\'e
first return map.
\item[(b)] Define what it means for the orbit \(\Gamma_0\) to be \textit{hyperbolic}.
Your definition should make the following theorem true.

\textbf{Theorem 1.} Suppose that the \(\Gamma_0\) is a hypererbolic nonconstant
periodic orbit of the system (3). Then there is an \(\eta>0\) depending on \(f\) and
\(\Gamma_0\) and a continuous one-parameter family of closed curves \(\Gamma_\epsilon\)
such that \(\Gamma_0=\Gamma_{\epsilon_0}\) and that \(\Gamma_\epsilon\) are nonconstant
periodic orbits of the perturbed systems
\[
  \frac{dx}{dt}=f(x,\epsilon)
\]
for all \(\epsilon\) such that \(\lvert\epsilon-\epsilon_0\rvert<\eta\).
\item[(c)] Sketch a proof of Theorem 1.
\item[(d)] Show that the \(\epsilon=1\) solution \((x=\cos t)\) can be continued to a one
parameter family of solutions in a neighberhood of \(\epsilon=1\) by checking that the
conditions of the theorem are satisfied:
\[
  \ddot{x}+(x^2+\dot{x}^2-\epsilon)\dot{x}+x=0.
\]
\end{enumerate}