$\omega$- and $\alpha$-limit sets I'm trying to identify the $\omega$-limit sets and $\alpha$-limit sets of $r' = r-r^2$, and $\theta' = 1$.
I note that the $\omega$-limit sets are essentially sinks and sources are $\alpha$-limit sets, but am confused on the procedure of such.
Also, how can we use this concept to tackle: $r' = r^3 - 3r^2 + 2r$, and $\theta' = 1$
 A: Consider the system:
$$r' = r(1-r), ~~\text{and}~~ \theta' = 1,$$
where $r \ge 0$.
The radial and angular dynamics are uncoupled and so can be analyzed separately.
Treating $r' = r(1-r)$ as a vector field on the line (plot of $r' vs. r$), we see that $r^*=0$ is an unstable fixed point and $r^*=1$ is stable (plot these).
If we now look at the phase plane, all trajectories (except $r^*=0$) approach the unit circle $r^* = 1$ monotonically. 
Since the motion in the $\theta$-direction is simply a rotation at constant angular velocity, we see that all trajectories spiral asymptotically toward a limit cycle at $r=1$.
Note, you can also plot solutions as a function of time, for example, you know, from polar coordinates, what $x(t)$ is. Start a trajectory outside the limit cycle and see if it settles down to the limit cycle.
You can now use this as a model to do the next one.
For the second one, you can write:
$$r' = r^3 - 3r^2 +2r = r(r^2-3r+2) = r(r-1)(r-2), ~~\theta' = 1.$$
Using the previous example, you should be able to analyze.
