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Symbolic Computation of an Arbitrary-Order Resonance Condition in a Hamiltonian System

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Abstract

The study of formal stability of equilibrium positions of a multiparametric Hamiltonian system in a generic case is traditionally carried out using its normal form under the condition of the absence of resonances of small orders. In this paper we propose a method of symbolic computation of the condition of existence of a resonance of arbitrary order for a system with three degrees of freedom. It is shown that this condition for each resonant vector can be represented as a rational algebraic curve. By methods of computer algebra the rational parametrization of this curve for the case of an arbitrary resonance is obtained. A model example of some two-parameter system of pendulum type is considered.

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ACKNOWLEDGMENTS

The authors express their gratitude to Professor A.D. Bruno for his useful remarks and discussion of this paper.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047, Moscow, Russia

    A. B. Batkhin

  2. Moscow Institute of Physics and Technology, Institutskii per. 9, 141701, Dolgoprudnyi, Moscow oblast, Russia

    A. B. Batkhin

  3. Sharof Rashidov Samarkand State University, Universitetskii bul’v. 15, 140104, Samarkand, Uzbekistan

    Z. Kh. Khaidarov

Authors
  1. A. B. Batkhin
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  2. Z. Kh. Khaidarov
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Correspondence to A. B. Batkhin or Z. Kh. Khaidarov.

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The authors declare that they have no conflicts of interest.

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Translated by Yu. Kornienko

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Batkhin, A.B., Khaidarov, Z.K. Symbolic Computation of an Arbitrary-Order Resonance Condition in a Hamiltonian System. Program Comput Soft 49, 842–853 (2023). https://doi.org/10.1134/S0361768823080030

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  • DOI: https://doi.org/10.1134/S0361768823080030

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