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Эта публикация цитируется в 9 научных статьях (всего в 9 статьях)
On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Case of Combinational Resonance
Boris S. Bardinab, Evgeniya A. Chekinaa a Department of Mechatronic and Theoretical Mechanics,
Faculty of Information Technologies and Applied Mathematics,
Moscow Aviation Institute (National Research University),
Volokolamskoe sh. 4, Moscow, 125993 Russia
b Computer Modelling Laboratory, Department of Mechanics and Control of Machines, Mechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN), M. Kharitonyevskiy per. 4, Moscow, 101990 Russia
Аннотация:
We deal with a Hamiltonian system with two degrees of freedom,
whose Hamiltonian is a 2$\pi$-periodic function of time and analytic
in a neighborhood of an equilibrium point. It is assumed that the
characteristic equation of the system linearized in a neighborhood
of the equilibrium point has two different double roots such that
their absolute values are equal to unity, i. e., a combinational
resonance takes place in this system. We consider the case of
general position when the monodromy matrix of the linearized system
is not diagonalizable. In this case the equilibrium point is linearly unstable. However,
this does not imply its instability in the original nonlinear system. Rigorous
conclusions on the stability can be formulated in terms of
coefficients of the Hamiltonian normal form.
We describe a constructive algorithm for constructing and
normalizing the symplectic map generated by the phase flow of the
Hamiltonian system considered. We obtain explicit relations between
the coefficients of the generating function of the symplectic map
and the coefficients of the Hamiltonian normal form. It allows us
to formulate conditions of stability and instability in terms of
coefficients of the above generating function. The developed
algorithm is applied to solve the stability problem for
oscillations of a satellite with plate mass geometry, that is, $J_z
= J_x +J_y$, where $J_x$, $J_y$, $J_z$ are the principal moments of
inertia of the satellite, when the parameter values belong to a boundary of linear stability.
Ключевые слова:
Hamiltonian system, stability, symplectic map, normal form, oscillations, satellite.
Поступила в редакцию: 14.12.2018 Принята в печать: 04.02.2019
Образец цитирования:
Boris S. Bardin, Evgeniya A. Chekina, “On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Case of Combinational Resonance”, Regul. Chaotic Dyn., 24:2 (2019), 127–144
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd449 https://www.mathnet.ru/rus/rcd/v24/i2/p127
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