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On the Stability of a Periodic Hamiltonian System with One Degree of Freedom in a Transcendental Case

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Abstract

The stability of an equilibrium of a nonautonomous Hamiltonian system with one degree of freedom whose Hamiltonian function depends 2π-periodically on time and is analytic near the equilibrium is considered. The multipliers of the system linearized around the equilibrium are assumed to be multiple and equal to 1 or–1. Sufficient conditions are found under which a transcendental case occurs, i.e., stability cannot be determined by analyzing the finite-power terms in the series expansion of the Hamiltonian about the equilibrium. The equilibrium is proved to be unstable in the transcendental case.

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), Moscow, 125993, Russia

    B. S. Bardin

  2. Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, 101990, Russia

    B. S. Bardin

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  1. B. S. Bardin
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Correspondence to B. S. Bardin.

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Original Russian Text © B.S. Bardin, 2018, published in Doklady Akademii Nauk, 2018, Vol. 479, No. 5, pp. 485–488.

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Bardin, B.S. On the Stability of a Periodic Hamiltonian System with One Degree of Freedom in a Transcendental Case. Dokl. Math. 97, 161–163 (2018). https://doi.org/10.1134/S1064562418020163

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  • DOI: https://doi.org/10.1134/S1064562418020163

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