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Zbl 1223.70054
Xu, Ming; Xu, Shijie
Nonlinear dynamical analysis for displaced orbits above a planet.
(English)
[J] Celest. Mech. Dyn. Astron. 102, No. 4, 327-353 (2008). ISSN 0923-2958; ISSN 1572-9478/e

Summary: 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 $\tau _{1}$ and $\tau _{2}$ directions, and a rough algorithm for calculating $\tau _{1}$ and $\tau _{2}$ 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.
MSC 2000:
*70F15 Celestial mechanics
70H08 Nearly integrable Hamiltonian systems, KAM theory

Keywords: displaced orbits; solar sail; nonlinear dynamics; bifurcation; quasiperiodic; homoclinic orbits; KAM torus; periodic Lyapunov orbits

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