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Regular and Chaotic Dynamics

, Volume 22, Issue 7, pp 808–823 | Cite as

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. BardinEmail author
  • Evgeniya A. Chekina
Article

Abstract

This paper is concerned with a nonautonomous Hamiltonian system with two degrees of freedom whose Hamiltonian is a 2π-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.

Keywords

Hamiltonian system stability symplectic map normal form resonant rotation satellite 

MSC2010 numbers

34D20 37J40 70K30 70K45 37N05 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mechatronics and Theoretical Mechanics, Faculty of Information Technologies and Applied MathematicsMoscow Aviation Institute (National Research University)MoscowRussia
  2. 2.Computer Modelling Laboratory, Department of Mechanics and Control of MachinesMechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN)MoscowRussia

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