Math 6410 • ODE

Consider the system
\[
  x'=f(x)
  \tag{4.1}
\]
where \(f\in C^1(\R^n,\R^n)\), and suppose that (4.1) has a periodic solution \(\gamma(t)\)
of least period \(T\).
\begin{enumerate}
\item Define what it means to say that \(\gamma(t)\) is orbitally asymptotically stable
with asymptotic phase.
\item Let
\[
  \xi'=A(t)\xi
  \tag{4.2}
\]
be the variational equation (the linearization of (4.1)) relative to \(\gamma(t)\).
State the Floquet representation for the fundamental matrix solution of (4.2) and
discuss the stability of \(\gamma(t)\) in terms of properties of the fundamental matrix
solution.
\item Consider the system
\[
\begin{aligned}
  x' &= x-y-x(x^2+y^2),\\
  y' &= x+y-y(x^2+y^2).
\end{aligned}
\]
Find a nontrivial periodic solution of this equation, and prove (in the easiest way
possible) that this periodic solution is orbitally asymptotically stable with
asymptotic phase. Compute the asymptotic phase explicitly.
\item Compute the Poincare map associated with the periodic orbit you found in (c).
\end{enumerate}