Multibump trajectories of adiabatically perturbed periodic Hamiltonian systems with pitchfork bifurcations

We study a $1\tfrac{1}{2}$-degrees of freedom Hamiltonian system with a potential $U(x,\varepsilon t) = \tfrac{1}{2}(\varphi (\varepsilon t)x^2 - x^4)$ 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.