Multibump trajectories of adiabatically perturbed periodic Hamiltonian systems with pitchfork bifurcations

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.