Nonlinear Systems & Linearization · Challenge problem

The system $x' = y,\; y' = -x + x^2$ has a saddle at $(1, 0)$ and a homoclinic orbit: a loop leaving the saddle and returning to it, surrounding the center at the origin. Using the conserved energy $E = \dfrac{y^2}{2} + \dfrac{x^2}{2} - \dfrac{x^3}{3}$, find the area enclosed by this loop.