I have the dynamical system
$\dot r = r(r^2-9)(r^2-16)^2 \\ \dot \theta = r^2 \\ \dot z = z(1-r^2)$
I am asked to compute the $\omega$-limit set for every point in $\Re^3$.
I have been given hints to divide $\Re^3$ into regions with similar $\omega$-limit sets and then give these sets for every point in such a region. And also, consider first the asymptotic behavior in the planar system $(r, z)$ and then put $\theta$ back in .
The phase flow for $(r,z)$ looks like this http://m.wolframalpha.com/input/?i=streamplot%5B%7Bx%28x%5E2-9%29%28x%5E2-16%29%5E2%2Cy%281-x%5E2%29%7D%5D
In the 2-D system of $(r,z)$ there are fixed points at $(0,0)$, $(3,0)$, $(-3,0)$, $(4,0)$, $(-4,0)$ but I am not to sure as to how this will help me
Edit: Can I say that $(r,z)=(0,0)$ will give me a fixed point of the complete system, hence that is a limit point?
Edit: It is clear from the system that the fixed points of the system have coordinates $(0,\theta,0)$, i.e. the whole $\theta$ axis are fixed points