Nonlinear Systems & Linearization · Superchallenge problem

The system $x' = -y + x^3,\; y' = x + 2y^3$ has linearization a center at the origin, so the nonlinear terms decide. A solution starting at $(s, 0)$ with small $s > 0$ next crosses the positive $x$-axis at $(P(s), 0)$, where $P(s) = s + cs^3 + O(s^5)$. Find $c$.