Quasienergy integral for canonical maps

Canonical (area-preserving) maps of the phase plane of action-angle variables whose coefficients do not depend explicitly on the number of mapping steps are considered. Just as the absence of an explicit time dependence of the coefficients of a canonical system of differential equations leads to energy conservation, such maps may have an integral of the motion – called a quasienergy integral. It is shown that such an integral can be constructed in the form of a series of analytic functions, a perturbation-theory series, and the superconvergent series of Kolmogorov–Arnol'd–Moser (KAM) theory. These series converge only in limited regions of the phase plane, and their sums have simple poles at fixed (resonance) points of the map. For a sufficiently small perturbation constant $g$, it is possible to find approximate regular expressions for the quasienergy near any given resonance with any finite accuracy in $g$. The regions of applicability of the obtained expressions overlap, and this makes it possible to construct at small $g$ an approximate phase portrait of the map on the complete phase plane.