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Nonlinear dynamical analysis for displaced orbits above a planet

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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 max z , 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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References

  • Arnold V.I., Kozlov V.V., Neishtadt A.I. (1997) Mathematical Aspects of Classical and Celestial Mechanics. 2nd edn. Springer–Verlag, NY

    MATH  Google Scholar 

  • Baoyin H., McInnes C. (2006) Solar Sail Halo Orbits at the Sun-Earth Artificial L1 Point. Celest. Mech. Dyn. Astron. 94: 155–171. doi:10.1007/s10569-005-4626-3

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Barden, B.T., Howell, K.C.: Application of dynamical systems theory to trajectory design for a libration point mission. Paper No. AIAA-96-3602-CP (1996)

  • Bookless J., McInnes C. (2006) Dynamics and control of displaced periodic orbits using solar-sail propulsion. J. Guid. Control Dyn. 29(3): 527–537. doi:10.2514/1.15655

    Article  Google Scholar 

  • Bookless, J., McInnes, C.: Control of Lagrange point orbits using solar sail propulsion. IAC-05-C1.6.03 (2006b)

  • Celletti A., Chierchia L. (2006) KAM tori for N-body problems: a brief history. Celest. Mech. Dyn. Astron. 95: 117–139. doi:10.1007/s10569-005-6215-x

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Conley C. (1963) Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16: 732–746. doi:10.1137/0116060

    Article  MathSciNet  Google Scholar 

  • Danckowicz H. (1994) Some special orbits in the two-body problem with radiation pressure. Celest. Mech. Dyn. Astron. 58(4): 353–370. doi:10.1007/BF00692010

    Article  ADS  Google Scholar 

  • Golubitsky M., Marsden J.E. (1983) The Morse lemma in infinite dimensions via singularity theory. SIAM J. Math. Anal. 14(6): 1037–1044. doi:10.1137/0514083

    Article  MATH  MathSciNet  Google Scholar 

  • Koon W.S., Lo M.W., Marsden J.E., Ross S.D. (2000) Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10(2): 427–469. doi:10.1063/1.166509

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Llibre J., Martinez R., Simó C. (1985) Transversality of the invariant manifolds associated to Lyapunov family of periodic orbits near L 2 in the restricted three-body problem. J. Differ. Equ. 58: 104–156. doi:10.1016/0022-0396(85)90024-5

    Article  MATH  Google Scholar 

  • McInnes C.R. (1999) Solar Sailing: Technology, Dynamics and Mission Applications. Springer–Verlag, London

    Google Scholar 

  • Meyer, K.R., Hall, R.: Hamiltonian Mechanics and the n-Body Problem. Springer-Verlag, Applied Mathematical Sciences (1992)

  • Moser J. (1958) On the generalization of a theorem of Liapunov. Commun. Pure Appl. Math. 11: 257–271. doi:10.1002/cpa.3160110208

    Article  MATH  Google Scholar 

  • Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Texts Appl. Math. Sci. 2, Springer–Verlag (1990)

  • Xu,Ming., Xu, Shijie. (2007) J 2 invariant relative orbits via differential correction algorithm. Acta Mech. Sin. 23(5): 585–595. doi:10.1007/s10409-007-0097-y

    Article  ADS  MathSciNet  Google Scholar 

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  1. School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China

    Ming Xu & Shijie Xu

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  1. Ming Xu
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  2. Shijie Xu
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Correspondence to Ming Xu.

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Xu, M., Xu, S. Nonlinear dynamical analysis for displaced orbits above a planet. Celest Mech Dyn Astr 102, 327–353 (2008). https://doi.org/10.1007/s10569-008-9171-4

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