Celestial Mechanics and Dynamical Astronomy

, Volume 102, Issue 4, pp 327–353

Nonlinear dynamical analysis for displaced orbits above a planet

Original Article

DOI: 10.1007/s10569-008-9171-4

Cite this article as:
Xu, M. & Xu, S. Celest Mech Dyn Astr (2008) 102: 327. doi:10.1007/s10569-008-9171-4


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 hzmax, 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.


Displaced orbitsSolar sailNonlinear dynamicsBifurcationQuasiperiodicHomoclinic orbitsKAM TorusPeriodic Lyapunov orbits

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of AstronauticsBeijing University of Aeronautics and AstronauticsBeijingChina