Is there an appropriate name for a set $C$ of points in a dynamical system with all the properties:
- $C$ is a closed set with the topology of $S^1$
- Trajectories within $C$ remain in $C$
- $C$ is the limit set of various other trajectories (i.e. given such a trajectory, it approaches all points on $C$ arbitrarily closely, even though it does not intersect $C$)
- $C$ contains a fixed point (not the interior of $C$, but $C$ itself)
I wanted to call this a limit cycle - except that it contains a fixed point and hence is not a cycle. Something like 'attractor' fails to be specific enough, as it does not capture the limit-cycle-like behaviour.
For a simple example, consider $(r,\theta)$ polar coordinates with $$\frac{\rm d}{{\rm d} s}(r,\theta) = (-(r-1)^3, \sin^2 \frac{\theta}{2} + (r-1)^2)$$ Clearly as $s\to \infty$, generically $r \to 1$. One can also check that $\theta$ wraps around infinitely many times. And the set $r=1$ is mapped to itself. But there is a fixed point at $(r,\theta)=(1,0)$ so $r=1$ is not a cycle.