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.
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References
V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983).
V. V. Kozlov, “Splitting of separatrices and the appearance of isolated periodic solutions in Hamiltonian systems with 1.5 degrees of freedom,” Usp. Mat. Nauk, 41, No. 5, 177–178 (1986).
A. Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “Buffering phenomenon in systems close to two-dimensional Hamiltonian systems,” Tr. Inst. Mat. Mekh. Ural Otdel. Ros. Akad. Nauk, 12, No. 1, 109–141 (2006).
S. D. Glyzin, A.Yu. Kolesov, and N. Kh. Rozov, “Buffering phenomenon in systems with 1.5 degrees of freedom,” Zh. Vychisl. Mat. Mat. Fiz., 46, No. 9, 1582–1593 (2006).
S. D. Glyzin, A.Yu. Kolesov, and N. Kh. Rozov, “On the limiting values of Mel’nikov functions on periodic orbits,” Differents. Uravn., 43, No. 2, 176–190 (2007).
Yu. E. Vakal and I. O. Parasyuk, “Estimation of the number of perturbed ultrasubharmonics for a system with 1.5 degrees of freedom close to a Hamiltonian system,” Nelin. Kolyv., 14, No. 2, 147–180 (2011); English translation: Nonlin. Oscillations, 14, No. 2, 149–186 (2011).
V. I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics [in Russian], Izhevsk (1999).
V. Guillemin and S. Sternberg, Geometric Asymptotics, American Mathematical Society, Providence, RI (1977).
H. L. Montgomery, “Fluctuations in the mean of Euler’s phi function,” Proc. Indian Acad. Sci. (Math. Sci.), 97, No. 1-3, 239–245 (1987).
A. Ya. Khinchin, Continued Fractions [in Russian], Nauka, Moscow (1978).
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings. Classification of Critical Points, Caustics, and Wave Fronts [in Russian], Nauka, Moscow (1982).
F. Dress, “Discrepance des suites de Farey,” J. Theor. Nombres Bordeaux, 11, No. 2, 345–367 (1999).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 4, pp. 463–489, April, 2012.
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Vakal, Y.E., Parasyuk, I.O. Estimation of the number of ultrasubharmonics for a two-dimensional almost autonomous Hamiltonian system periodic in time. Ukr Math J 64, 525–554 (2012). https://doi.org/10.1007/s11253-012-0663-8
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DOI: https://doi.org/10.1007/s11253-012-0663-8