 Solution

A form of the Poincaré-Bendixson theorem states that a nonempty omega-limit set of a forward orbit of a $C^1$ planar autonomous vector field, contained in a compact region and containing no equilibrium, is a periodic orbit.

Set $\rho =x^2+y^2$. Direct calculation givesbecause $0\le x^2{y}^2\le {\rho }^2/4$. At the inner boundary $\rho =2/(2+k^2)$ the flow points into or along the annulus, and at the outer boundary $\rho =1$ it does likewise. Smoothness and these inequalities make the compact annulus forward invariant.

For nonzero $(x,y)$, write $x=\sqrt{\rho }\cos \theta$, $y=\sqrt{\rho }\sin \theta$. Thenthroughout the annulus. Hence it contains no equilibrium. Any orbit starting there remains compactly trapped, has a nonempty omega-limit set, and the Poincaré-Bendixson theorem supplies the required periodic orbit in the stated annulus. For $k=0$ the annulus reduces to $\rho =1$ itself, and there $\dot{\theta }=1$, giving the periodic orbit directly.