Nonlinear Systems & Linearization · Superchallenge problem

For $x' = x - y - x(x^2 + y^2),\; y' = x + y - y(x^2 + y^2)$, let $P(s)$ be the first-return (Poincaré) map of the positive $x$-axis: a solution starting at $(s, 0)$ with $s > 0$ next returns to the positive $x$-axis at $(P(s), 0)$. Find $P'(1)$.