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Эта публикация цитируется в 8 научных статьях (всего в 8 статьях)
On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Second-order Resonance Case
Boris S. Bardinab, Evgeniya A. Chekinaa a Department of Mechatronics 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:
This paper is concerned with a nonautonomous 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 system exhibits a secondorder resonance, i. e., the system linearized in a neighborhood of the equilibrium point has a double multiplier equal to $-1$. The case of general position is considered when the monodromy matrix is not reduced to diagonal form and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system.
In this paper, a constructive algorithm for a rigorous stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed in [1]. The main idea of this method is to construct and normalize a symplectic map generated by the phase flow of a Hamiltonian system.
It is shown that the normal form of the Hamiltonian function and the generating function of the corresponding symplectic map contain no third-degree terms. Explicit formulae are obtained which allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of a symplectic map.
The developed algorithm is applied to solve the problem of stability of resonant rotations of a symmetric satellite.
Ключевые слова:
Hamiltonian system, stability, symplectic map, normal form, resonant rotation, satellite.
Поступила в редакцию: 03.08.2017 Принята в печать: 19.10.2017
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 Second-order Resonance Case”, Regul. Chaotic Dyn., 22:7 (2017), 808–823
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd292 https://www.mathnet.ru/rus/rcd/v22/i7/p808
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