Consider the 2-dimensional flow

$$\dot{x}=y+\frac{1}{4} x\left(1-2 x^{2}-2 y^{2}\right), \quad \dot{y}=-x+\frac{1}{2} y\left(1-x^{2}-y^{2}\right)$$

Use the Poincaré-Bendixson theorem, which should be stated carefully, to obtain a domain $\mathcal{D}$ in the $x y$-plane, within which there is at least one periodic orbit.