Math 6410 • ODE

Suppose that \(f,g:\R^3\to\R\) are smooth functions. Consider the parameter dependent
system
\begin{equation}
\begin{aligned}
  \dot{x} &= f(x,y;\epsilon),\\
  \dot{y} &= g(x,y;\epsilon).
\end{aligned}
\tag{4}
\end{equation}
\begin{enumerate}
\item[(a)] Suppose \(\Gamma_0\) is a nonconstant \(T\)-periodic solution of (4) when
\(\epsilon=0\). Define \textit{Poincar\`e Section}, \textit{Poincar\'e Map} for
\(\Gamma_0\).
\item[(b)] Find a nontrivial periodic solution \(\Gamma_0\) for the system (5) when
\(\epsilon=0\). Show that a nontrivial periodic solution \(\Gamma_\epsilon\) for the
\(\epsilon\)-system can be continued from \(\Gamma_0\) as the parameter varies through
small \(\lvert\epsilon\rvert\).
\begin{equation}
\begin{aligned}
  \dot{x} &= y+x(1-x^2-y^2),\\
  \dot{y} &= -x+y(1-x^2-y^2)+\epsilon y.
\end{aligned}
\tag{5}
\end{equation}
\end{enumerate}