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

Abstract

Using the Arnold method of detection of fixed points of symplectic diffeomorphisms, we find lower estimates for the number of ultrasubharmonics in a Hamiltonian system on a two-dimensional symplectic manifold with almost autonomous time-periodic Hamiltonian. We show that the asymptotic behavior of these estimates as the perturbation parameter tends to zero depends on which of the four zones of a ring domain foliated by closed level curves of the unperturbed Hamiltonian the generating unperturbed ultrasubharmonics belong to.