Oscillatory motions and stability problem of solutions to Hamiltonian systems with many degrees of freedom

The paper is devoted to qualitative study of oscillatory motions to Hamiltonian systems with the five dimensional phase space, which periodically depend on independent variable. We prove that an open set of initial data in the phase space generates oscillatory motions, which are stable in the Ljapunov sense. The special case of such systems is the classical model of accelerator of charged particles moving in varying periodic electric field and in constant magnetic field, and constructed oscillatory motions lead to unbounded growth of energy particles. We establish that the property of stability of solutions is preserved under small perturbation of the Hamiltonian function.