Math 6410 • ODE

\begin{enumerate}

\item
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
\(\mathbb{R}^n\), and determine it in the case of a trajectory for which
\(|z(0)|<1\).


\item
Use the Poincare-Bendixson Theorem and the fact that the planar system
\[
\dot{x}=x-y-x^3,\quad \dot{y}=x+y-y^3
\]
has only the one critical point at the origin to show that this system has a
periodic orbit in the annular region
\(A=\{x\in\mathbb{R}^2\mid 1<|x|<\sqrt{2}\}\).


\item
Give the definitions of Poincare and Liapunov stability. Show that solutions of the
system \(\dot{x}=y\), \(\dot{y}=0\) are Poincare but not Liapunov stable.


\item
Consider the system
\[
\dot{x}=x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}}
\]
\[
\dot{y}=x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}}.
\]
Show that the above pair of equations can be rewritten in polar coordinates as
\[
\dot{r}=r(1-r^2),\quad \dot{\theta}=2\sin^2(\theta/2)
\]
and sketch the phase portrait. Determine whether or not the fixed point (1, 0) is
Liapunov stable.

\end{enumerate}