= Exponential radial approach to a unit-circle limit cycle
{title2=$\dot r=1-r,\quad\dot\theta=1$}
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>.
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