Nonlinear Systems & Linearization · Superchallenge problem

The system $x' = -\dfrac{y}{2} + x\left(1 - x^2 - \dfrac{y^2}{4}\right),\; y' = 2x + y\left(1 - x^2 - \dfrac{y^2}{4}\right)$ has exactly one closed orbit. Find the area it encloses.