Symbolic computation of an arbitrary-order resonance condition in a Hamiltonian system

A. B. Batkhin^ab, Z. Kh. Khaydarov<sup>c

a</sup> Keldysh Institute of Applied Mathematics of Russian Academy of Sciences
^b Moscow Institute of Physics and Technology
^c Samarkand State University

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.

Keywords: Hamiltonian system, equilibrium state, normal form, formal stability, resonance condition, elimination ideal

DOI: 10.15514/ISPRAS-2023-35(4)-12