= Unit-circle attracting limit cycle
{title2=$\dot r=r(1-r^2),\quad\dot\theta=-1$}
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 $r(t_0)=r_0>0$,
$$
r^2(t)=\frac1{1+(r_0^{-2}-1)e^{-2(t-t_0)}}.
$$
Interior <orbits> approach the origin backward in time, but exterior <orbits> have backward <finite-time blow-up of an ordinary differential equation> at $t=t_0+\tfrac12\log(1-r_0^{-2})$. Thus forward convergence to a <limit cycle> does not imply existence for all negative times.
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