We study the complex dynamics in analytic area-preserving maps in a neighbourhood of a resonant elliptic fixed point. We assume that the resonance is weak, i.e., the linear part is a rotation of an angle 2\pi where n\geq5. Normal form theory suggests that there is a flower with n petals which consists of points bi-asymptotic to the fixed point.
We show that the flower splits: there are parabolic stable and unstable complex manifolds and they do not intersect. We measure the splitting of the manifolds and relate it to the Stokes phenomenon.