Math 6410 • ODE

\begin{enumerate}
\item[(a)] Give definitions for the following: \textit{invariant set}, \textit{attracting set},
\textit{\(\omega\)-limit set}.
\item[(b)] Determine the invariant sets and the attracting set of the dynamical system
\[
\begin{aligned}
  \dot{x} &= -y+x(1-x^2-y^2),\\
  \dot{y} &= x+y(1-x^2-y^2),\\
  \dot{z} &= \alpha>0.
\end{aligned}
\]
Also sketch the flow.
\item[(c)] Describe what happens to the flow if we identify the points \((x,y,0)\) and
\((x,y,2\pi)\) in the planes \(z=0\) and \(z=2\pi\).
\item[(d)] By explicitly constructing solutions on the invariant torus
\(x^2+y^2=1\), \(0\le z<2\pi\), show that the torus is only an attractor if \(\alpha\)
is irrational.
\end{enumerate}