Nonlinear Systems & Linearization · Challenge problem

Consider $x' = x - y - x(x^2 + 2y^2),\; y' = x + y - y(x^2 + 2y^2)$. Let $a$ be the largest radius such that $r' \ge 0$ at every point of the circle $x^2 + y^2 = a^2$, and $b$ the smallest radius such that $r' \le 0$ at every point of the circle $x^2 + y^2 = b^2$. The annulus $a \le r \le b$ is then a trapping region containing a limit cycle. Find its area.