We study a Hamiltonian system with 1.5 degrees of freedom with a Hamiltonian slowly varying in time. The bifurcation parameter of the system is a twice continuously differentiable periodic function with simple zeros. The presence of zeros of the function implies existence of a cascade of pitchfork bifurcations in the phase space of the "frozen"system. In this work we obtain a sufficient condition for the existence of transversal "rotating"homoclinic trajectories of the system. The result is based on asymptotic analysis of solutions to the Cauchy problem in different domains of the phase space.