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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 1, Pages 63–73 (Mi rcd61)

On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit

Boris S. Bardin, Evgeniya A. Chekina, Alexander M. Chekin

Theoretical Mechanics Department, Faculty of Applied Mathematics and Physics, Moscow Aviation Institute, Volokolamskoe sh. 4, Moscow, 125871, Russia

Abstract: We study the Lyapunov stability problem of the resonant rotation of a rigid body satellite about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the satellite completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the above resonance rotation was started in [4, 6]. In the present paper, rigorous stability conclusions in the previously unstudied range of parameter values are obtained. In particular, new intervals of stability are found for eccentricity values close to 1. In addition, some special cases are studied where the stability analysis should take into account terms of degree not less than six in the expansion of the Hamiltonian of the perturbed motion. Using the technique described in [7, 8], explicit formulae are obtained, allowing one to verify the stability criterion of a time-periodic Hamiltonian system with one degree of freedom in the special cases mentioned.

Keywords: Hamiltonian system, symplectic map, normal form, resonance, satellite, stability

 Funding Agency Grant Number Russian Science Foundation 14-21-00068 The work was carried out under the grant of the Russian Scientific Foundation (project No 14-21-00068) at the Moscow Aviation Institute (National Research University).

DOI: https://doi.org/10.1134/S1560354715010050

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MSC: 34C15, 34C20, 34C23, 34C25
Accepted:13.12.2014
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Citation: Boris S. Bardin, Evgeniya A. Chekina, Alexander M. Chekin, “On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit”, Regul. Chaotic Dyn., 20:1 (2015), 63–73

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. S. Bardin, E. A. Chekina, “Ob ustoichivosti rezonansnogo vrascheniya sputnika na ellipticheskoi orbite”, Nelineinaya dinam., 12:4 (2016), 619–632
2. Boris S. Bardin, Evgeniya A. Chekina, “On the Stability of Resonant Rotation of a Symmetric Satellite in an Elliptical Orbit”, Regul. Chaotic Dyn., 21:4 (2016), 377–389
3. 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
4. Tatyana E. Churkina, Sergey Y. Stepanov, “On the Stability of Periodic Mercury-type Rotations”, Regul. Chaotic Dyn., 22:7 (2017), 851–864
5. J. Chu, Z. Liang, P. J. Torres, Zh. Zhou, “Existence and stability of periodic oscillations of a rigid Dumbbell satellite around its center of mass”, Discrete Contin. Dyn. Syst.-Ser. B, 22:7 (2017), 2669–2685
6. B. S. Bardin, E. A. Chekina, “On the constructive algorithm for stability investigation of an equilibrium point of a periodic Hamiltonian system with two degrees of freedom in first-order resonance case”, Mech. Sol., 53:2 (2018), S15–S25
7. B. S. Bardin, A. N. Avdushkin, “Stability analysis of an equilibrium position in the photogravitational Sitnikov problem”, Eighth Polyakhov's Reading, AIP Conf. Proc., 1959, eds. E. Kustova, G. Leonov, N. Morosov, M. Yushkov, M. Mekhonoshina, Amer. Inst. Phys., 2018, 040002
8. B. S. Bardin, E. A. Chekina, “On orbital stability of planar oscillations of a satellite in a circular orbit on the boundary of the parametric resonance”, Eighth Polyakhov's Reading, AIP Conf. Proc., 1959, eds. E. Kustova, G. Leonov, N. Morosov, M. Yushkov, M. Mekhonoshina, Amer. Inst. Phys., 2018, 040003
9. Z. Liang, F. Liao, “Periodic solutions for a dumbbell satellite equation”, Nonlinear Dyn., 95:3 (2019), 2469–2476