Estimation of the number of ultrasubharmonics for a two-dimensional almost autonomous Hamiltonian system periodic in time

The Arnold method for the detection of fixed points of symplectic diffeomorphisms is used to establish lower estimates for the number of ultrasubharmonics in a Hamiltonian system on a two-dimensional symplectic manifold with an almost autonomous Hamiltonian periodic in time. It is shown that the asymptotic behavior of these estimates (as the small parameter of perturbation tends to zero) depends on the zone (from the set four zones of an annular domain foliated by the closed level curves of the unperturbed Hamiltonian) containing the generating unperturbed ultrasubharmonics.