Math 6410 • ODE

\begin{enumerate}
\item[(a)] Show how to convert the ODE
\[
  \dot{x}=f_1(x,y),\qquad \dot{y}=f_2(x,y).
\]
into polar coordinates.
\item[(b)] Consider the dynamical system
\[
\begin{aligned}
  \dot{x} &= -y+x(1-z^2-x^2-y^2),\\
  \dot{y} &= x+y(1-z^2-x^2-y^2),\\
  \dot{z} &= 0.
\end{aligned}
\]
Determine the invariant sets and attracting set of the system. Give a general
definition of the \(\omega\)-limit set for a flow \(\phi(x,t)\) in \(\R^n\), and determine
it in the case of a trajectory for which \(\abs{z(0)}<1\).
\item[(c)] Use the Poincare-Bendixson (PB) Theorem and the fact that the planar system
\[
  \dot{x}=x-y-x^3,\qquad \dot{y}=x+y-y^3
\]
has only the one critical point at the origin to show that this system has a periodic
orbit in the annular region \(A=\{x\in\R^2\mid 1<\abs{x}<\sqrt{2}\}\).
\end{enumerate}