On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance

2017 ◽  
Vol 22 (7) ◽  
pp. 773-781 ◽  
Author(s):  
Anatoly P. Markeev
Keyword(s):  
Hamiltonian System ◽  
Fourth Order ◽  
Critical Case ◽  
Periodic Motions ◽  
Order Resonance ◽  
Autonomous Hamiltonian System ◽  
The Stability

scholarly journals On the Stability of Periodic Motions of a Two-Body SystemWith Flexible Connection in an Elliptical Orbit

2021 ◽  
Author(s):  
Xue Zhong ◽  
Jie Zhao ◽  
Kaiping Yu ◽  
Minqiang Xu
Keyword(s):  
Periodic Solutions ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Elliptical Orbit ◽  
Periodic Motions ◽  
Nonlinear Stability Analysis ◽  
The Third ◽  
The Stability ◽  
Lyapunov Instability

Abstract This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third and fourth order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coeffcients of the normalized Hamiltonian. Then a nonlinear stability analysis for two periodic solutions is performed in the third and fourth order resonance cases as well as in the nonresonance case.

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