Dynamics of slow-fast Hamiltonian systems

Slow-fast Hamiltonian systems appear in many applications, in particular in the problem of Arnold's diffusion. When the slow variables are fixed we obtain the frozen system. If the frozen system has one degree of freedom and the level curves of the frozen Hamiltonian are closed, there is an adiabatic invariant which governs evolution of the slow variables. Near a separatrix of the frozen system the adiabatic invariant is destroyed. A.Neishtadt proved that at a crossing of the separatrix the adiabatic invariant has "random" jumps and the slow variables evolve in a quasi-random way. In this talk we discuss partial extension of Neishtadt's results to multidimensional slow-fast systems. The slow variables shadow trajectories of an effective Hamiltonian system which depends on a "random" integer parameter.