Math 6410 • ODE

Consider a nonconstant \(T\)-periodic orbit \(\gamma(t)\) of the smooth autonomous
differential equation
\[
  \dot{x}=f(x),\qquad x\in\R^n.
\]
\begin{enumerate}
\item[(a)] Describe the Poincar\'e Return Map associated to \(\gamma\).
\item[(b)] A \(2\pi\)-periodic solution of the planar system
\[
\begin{aligned}
  \dot{x} &= x-(x+y)\sqrt{x^2+y^2},\\
  \dot{y} &= y+(x-y)\sqrt{x^2+y^2}
\end{aligned}
\]
is given by \(\gamma(t)=(\cos t,\sin t)\). Find a Poincar\'e Return Map associated
to \(\gamma\).
\item[(c)] Find the derivative of the Poincar\'e Return Map from part (b) at \(\gamma\).
\item[(d)] Using (c), determine whether the solution (b) is stable and define the
sense of stability you mean.
\end{enumerate}