Math 6410 • ODE

Consider the system of equations

\[
\frac{dx}{dt}=\cos^2(x)\bigl(y\cos(x)-\sin(x)\bigr),\quad
\frac{dy}{dt}=-\sin(x)-y\cos(x),
\]

with \(-\frac{\pi}{2}\le x\le \frac{\pi}{2}\).
\begin{enumerate}

\item
Sketch a phase portrait for this system of equations. Identify all fixed points and their
stability.


\item
Show that the rectangle with vertices \((x_0,y_0)\), \((-x_0,y_0)\),
\((-x_0,-y_0)\), \((x_0,-y_0)\), where \(y_0=\tan(x_0)\), is positively invariant
for any \(x_0\), \(0<x_0<\frac{\pi}{2}\).


\item
What is \(\omega_+(x,y)\) for \(-\frac{\pi}{2}<x<\frac{\pi}{2}\)? Are
there any periodic orbits?


\item
What is \(\omega_+(x,y)\) for \(x=\frac{\pi}{2}\)? What is \(\omega_+(x,y)\) for
\(x=-\frac{\pi}{2}\)?

\end{enumerate}