Multidimensional Hamiltonian systems: non-integrability and diffusion

Hamiltonian systems of differential equations that are little different from completely integrable systems are under consideration. If such a system is integrable, then the action variables cannot change strongly, and there is no diffusion. Thus the non-integrable behaviour of a Hamiltonian system is closely linked with the diffusion of slow variables. This range of problems is discussed for a subclass of Hamiltonian systems. Using this example a new mechanism of diffusion, different from the ‘standard’ scheme of transition chains, is considered. This mechanism is related to the breakdown of a large number of invariant tori with almost resonance sets of frequencies of the non-perturbed problem. On the formal side, this phenomenon is based on the non-boundedness of integrals of conditionally-periodic functions of time with zero mean value.