Unit-circle attracting limit cycle ($\dot{r}=r(1-r^2),\dot{\theta }=-1$)

The planar system $\dot{x}=y+(1-x^2-y^2)x$, $\dot{y}=-x+(1-x^2-y^2)y$ has polar coordinates $\dot{r}=r(1-r^2)$, $\dot{\theta }=-1$. Every nonzero orbit approaches the clockwise unit-circle limit cycle, of period $2\pi$. For $r(t_0)=r_0>0$,Interior orbits approach the origin backward in time, but exterior orbits have backward finite-time blow-up of an ordinary differential equation at $t=t_0+\frac{1}{2}\log (1-r_0^{-2})$. Thus forward convergence to a limit cycle does not imply existence for all negative times.

On the punctured plane, the vector field $\dot{x}=(1/r-1)x-y$, $\dot{y}=(1/r-1)y+x$ has the displayed polar equations. Its exact trajectories satisfy $r(t)=1+(r(0)-1)e^{-t}$ and $\theta (t)=\theta (0)+t$. Every forward trajectory approaches the unit circle, rotating counterclockwise, from inside or outside according to its initial radius. The circle is an attracting limit cycle with least period $2\pi$. The origin is not an equilibrium: the original vector field is undefined there. The singularity prevents an application of a whole-plane Bendixson-Dulac criterion to this periodic orbit.