Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps

When the saddle-center bifurcation occurs in an analytic family of area-preserving maps, a parabolic fixed point first appears at the origin and then this point bifurcates, creating an elliptic fixed point and a hyperbolic one. Separatrices of the hyperbolic fixed point form a small loop around the elliptic point. In general the separatrices intersect transversely and the splitting is exponentially small with respect to the perturbation parameter. We derive an asymptotic formula, which describes the splitting, and study the properties of the pre-exponential factor.