Quasienergy integral for canonical maps
V. V. Sokolov
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.
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Theoretical and Mathematical Physics, 1986, 67:2, 464–473
V. V. Sokolov, “Quasienergy integral for canonical maps”, TMF, 67:2 (1986), 223–236; Theoret. and Math. Phys., 67:2 (1986), 464–473
Citation in format AMSBIB
\paper Quasienergy integral for canonical maps
\jour Theoret. and Math. Phys.
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