Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 7D / 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 givesFor $r>0$, $|xy|\le r^2/2$, so $-9/8\le \dot{\theta }\le -7/8$. In particular 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 annulusis 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 at least one periodic orbit in this annulus.