On Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes

The problem of averaging for systems with one fast phase was considered from various points of view in many papers. The averaging method of Krylov and Bogolyubov and methods of KAM theory originated this line of research, the most complete results were obtained by Neishtadt, where the coefficients are assumed real analytic. However, in many problems which are interesting from the point of view of applications, analytic dependence ceases to hold in the neighborhoods of some points. We show that one can choose a transformation such that the averaging procedure of Neishtadt is applicable in these neighborhoods and such passage to limit is uniform.