Nonlinear dynamical analysis for displaced orbits above a planet |
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Authors: | Ming Xu Shijie Xu |
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Institution: | (1) School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China |
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Abstract: | Nonlinear dynamical analysis and the control problem for a displaced orbit above a planet are discussed. It is indicated that
there are two equilibria for the system, one hyperbolic (saddle) and one elliptic (center), except for the degenerate h
z
max, a saddle-node bifurcation point. Motions near the equilibria for the nonresonance case are investigated by means of the
Birkhoff normal form and dynamical system techniques. The Kolmogorov–Arnold–Moser (KAM) torus filled with quasiperiodic trajectories
is measured in the τ 1 and τ 2 directions, and a rough algorithm for calculating τ 1 and τ 2 is proposed. A general iterative algorithm to generate periodic Lyapunov orbits is also presented. Transitions in the neck
region are demonstrated, respectively, in the nonresonance, resonance, and degradation cases. One of the important contributions
of the paper is to derive necessary and sufficiency conditions for stability of the motion near the equilibria. Another contribution
is to demonstrate numerically that the critical KAM torus of nontransition is filled with the (1,1)-homoclinic orbits of the
Lyapunov orbit. |
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Keywords: | Displaced orbits Solar sail Nonlinear dynamics Bifurcation Quasiperiodic Homoclinic orbits KAM Torus Periodic Lyapunov orbits |
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