We study a 1/2/-degrees of freedom Hamiltonian system with a potential U(x, εt) = 1/2(φ(εt)x 2 - x 4 ) slowly varying with time. It is assumed that the factor φ(τ) is a periodic function with simple zeroes on its period. Using WKB-method together with a modification of the Melnikov method, we prove that in the adiabatic limit a cascade of bifurcations, occuring when the factor φ passes through the zero value, leads to the existence of transversal homoclinic intersections and multibump trajectories of the system.