Institute of Mathematics of NAS of Ukraine,

3 Tereshchenkivs'ka Str.,

01601 Kyiv,

UKRAINE

and

Faculty of Applied Mathematics,

AGH University of Science and Technology,

30 Mickiewicz Al. Bl. A4,

30059 Krakow,

POLAND

E-mail: yarchyk@imath.kiev.ua

**V. Mel'nikov - A. Samoilenko adiabatic stability problem**

**Abstract:**

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.