V. Mel'nikov - A. Samoilenko adiabatic stability problem
We developed a new approach to the V. Mel'nikov - A. Samoilenko problem of the invariant torus stability based on the special construction of the so called "virtual" canonical transformations of the phase space of the system in the Hamilton-Jakobi variables. By means of these transformations the initial adiabatically perturbed oscillation type hamiltonian system takes the form of the hamiltonian system in N.N. Bogolyubov's canonical form and it makes it possible to use standard KAM theory scheme. In particular, the stability of the invariant torus of this system is found which solves the V. Mel'nikov - A. Samoilenko problem.