This question arose from a Dynamical Systems lecture I just came from. The professor constructed a system $\dot{r} = 1-r$, $\dot{\theta} = 1-r$, and showed that as $r(t)$ converges to $1$, $\theta(t)$ also converges to some finite $\theta_*$ (in particular, to $\theta_* = 1-r(0)+\theta(0)$). The phase portrait in $(x, y)$ is below.
However, for $\dot{r} = (1-r)^3$, $\dot{\theta} = 1-r$, in which $r(t)$ converges much more slowly to $1$, we have that $\theta(t)$ does not converge. Again, the phase portrait in $(x, y)$ is below.
It occurred to me that if we take the system $\dot{r} = \mathrm{sgn}(1-r)\lvert 1-r\rvert^m$, $\dot{\theta} = 1-r$, there is some $m = \inf\{m : \theta(t)\text{ does not converge}\} \in [1, 3]$. With a little work, we can see that this is $m = 2$ (in particular, just use separation of variables to solve $\dot{r} = -(r-1)^m$ for $r > 1$ and plug into $\dot{\theta}$, which we then see is not integrable for $m \geq 2$).
I worked this out for the professor and asked whether this event was classifiable as a bifurcation. He didn't seem to think so, but he also hadn't considered the problem in any depth. It seems to me that the limit cycle is stable for all $m \geq 1$, but the individual points of the limit cycle lose asymptotic stability. This doesn't seem to match the description of any type of bifurcation listed here. Does anybody have an idea of how to classify the behavior of this system around $m = 2$?