= Solution
A useful version of the <Poincaré-Bendixson theorem> says that a forward orbit of a continuously differentiable planar autonomous vector field which stays in a compact set has a nonempty compact omega-limit set; if that omega-limit set contains no equilibrium, it is a <periodic orbit>.
Let $r^2=x^2+y^2$. Direct calculation gives
$$
r\dot r=\frac14x^2(1-2r^2)+\frac12y^2(1-r^2),\qquad
\dot\theta=-1+\frac{xy}{4r^2}.
$$
For $r>0$, $|xy|\le r^2/2$, so $-9/8\le\dot\theta\le-7/8$. In particular \b[there is no equilibrium away from the origin].
On $r=1/2$, both coefficients in the radial expression are strictly positive, so the flow crosses outward. On $r=2$, both are strictly negative, so the flow crosses inward. Consequently the compact annulus
$$
\boxed{D=\{(x,y):1/4\le x^2+y^2\le4\}}
$$
is forward invariant. Any orbit starting there stays there, and its omega-limit set avoids the only equilibrium, the origin. The <Poincaré-Bendixson theorem> therefore supplies \b[at least one <periodic orbit> in this annulus].
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