Math 6410 • ODE

Let the system \(x'=f(x)\) define a \(C^1\) flow \(\phi_t\) on the open
set \(E\) contained in \(\R^n\). Prove that the positive limit set \(\Gamma^+(v)\) of a
trajectory \(x(t)\) with \(x(0)=v\) is closed. Then, select with a check-mark
\checkedchoice{} and solve either A-3.I or A-3.II:

\choicebox{} \textbf{A-3.I.} Consider the autonomous planar dynamical system
\[
  x'=6x-2xy-6x^2, \qquad y'=-7y+2xy-y^2.
\]
\begin{enumerate}
\item[(a)] Compute the four rest points (=equilibrium or stationary points) of the
system and the linearization about each rest point.
\item[(b)] Make a table in which each row contains a rest point, the classification
stable or unstable, and the geometric classification node, spiral, center or hyperbolic
point.
\item[(c)] Sketch the phase diagram showing the rest points and the local behavior of
solution curves (rough and brief!).
\end{enumerate}

\choicebox{} \textbf{A-3.II.} Let \(r^2=x^2+y^2\), \(w=(r^2-1)(r^2-4)\) and consider the
planar system \(x'=-y+xw\), \(y'=x+yw\) (\(r'=rw\), \(\theta'=1\) in polar coordinates). Apply
the Poincar\'e-Bendixson theorem to prove that \(r=1\) and \(r=2\) are limit cycles (a
periodic orbit \(\gamma\) with \(\gamma=\Gamma^+(v)\) or \(\gamma=\Gamma^-(v)\) for nearby
\(v\)).