Exponentially small splitting of separatrices in a weakly hyperbolic case

We validate the Poincaré–Melnikov method in the singular case of high-frequency periodic perturbations of the Hamiltonian h0(x,y)=(1/2)y2-x3+x4 under appropriate conditions, which among other things, imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one. The splitting is exponentially small.