Averaging Principle for Deterministic and Stochastic Perturbations I

First, I will consider averaging in the simplest case: for systems with one degree of freedom and the first integral without singularities. Then I will introduse various regularizations for the system with one degree of freedom and saddle points and show that in the general situation one should consider random perturbations of the equation, not just the initial condition, to regularize the averaging principle for deterministic perturbations. I will describe the limiting slow motion as a stochastic process on the corresponding graph. Next, I will give conditions for averaging principle to be valid for perturbations of multifrequency systems in the action-angle coordinates. I will consider some many-degrees of-freedom systems with singularities in the first integrals. In this case, an open book space should be considered as the phase space for the limiting slow motion. I will consider some applications of this theory to dynamics of incompressible fluid.