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Applications of lagrangian coherent structures to expression of invariant manifolds in astrodynamics

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Abstract

This paper investigates the relationship between invariant manifold and Lagrangian coherent structure (LCS) in dynamical systems. LCS is defined as the ridge of finite-time Lyapunov exponent (FTLE) field, and is proving to be excellent platform for studies of stable and unstable manifold in flows with arbitrary time dependence. In this study, the LCS tool is applied to autonomous systems, simple pendulum and planar circular restricted three-body problem (PCR3BP), and also non-autonomous ones, double-gyre flow and bicircular problem (BCP). A comparison between LCS and invariant manifold is presented.

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References

  • Barden, B.T., Howell, K.C., Lo, M.W.: Application of dynamical systems theory to trajectory design for a libration point mission. J. Astronaut. Sci. 45(2), 161–178 (1997)

    MathSciNet  Google Scholar 

  • Belbruno, E., Miller, J.: Sun-perturbed Earth-to-Moon transfers with ballistic capture. J. Guid. Control Dyn. 16(4), 770–775 (1993)

    Article  ADS  Google Scholar 

  • Dellnitz, M., Padberg, K., Post, M., Thiere, B.: Set oriented approximation of invariant manifolds: review of concepts for astrodynamical problems. In: New Trends in Astrodynamics and Applications III. AIP Conference Proceedings, vol. 886, pp. 90–99 (2007)

    Google Scholar 

  • Gawlik, E.S., Du Toit, P.C., Campagnola, S., Marsden, J.E.: Lagrangian coherent structures in the planar elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 103(3), 227–249 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Gong, S.P., Li, J.F., Baoyin, H.X.: Solar sail three-body transfer trajectory design. J. Guid. Control Dyn. 33(3), 873–886 (2010)

    Article  ADS  Google Scholar 

  • Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248–277 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Low energy transfer to the Moon. Celest. Mech. Dyn. Astron. 81(1–2), 63–73 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Li, M.T., Zheng, J.H.: Indirect transfer to the Earth–Moon L1 libration point. Celest. Mech. Dyn. Astron. 108(2), 203–213 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Lo, M.W., Williams, B.G., Bollman, W.E., Han, D., Hahn, Y., Bell, J.L., Hirst, E.A., Corwin, R.A., Hong, P.E., Howell, K.C., Barden, B., Wilson, R.: Genesis mission design. J. Astronaut. Sci. 49, 169–184 (2001)

    Google Scholar 

  • Parker, J.S., Anderson, R.L.: Targeting low-energy transfers to low lunar orbit. Acta Astronaut. 84(3–4), 1–14 (2013)

    Article  ADS  Google Scholar 

  • Peng, H.J., Zhao, J., Wu, Z.G., Zhong, W.X.: Optimal periodic controller for formation flying on libration point orbits. Acta Astronaut. 69(7–8), 537–550 (2011)

    Article  ADS  Google Scholar 

  • Qi, R., Xu, S.J., Xu, M.: Impulsive control for formation flight about libration points. J. Guid. Control Dyn. 35(2), 484–496 (2012a)

    Article  ADS  MathSciNet  Google Scholar 

  • Qi, R., Xu, S.J., Zhang, Y., Wang, Y.: Earth-to-Moon low energy transfer using time-dependent invariant manifolds. In: AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, Minnesota (2012b)

    Google Scholar 

  • Schoenmaekers, J., Horas, D., Pulido, J.A.: SMART-1: with solar electric propulsion to the Moon. In: 16th International Symposium on Space Flight Dynamics, Pasadena, California (2001)

    Google Scholar 

  • Short, C.R., Howell, K.C.: Lagrangian coherent structures in various map representations for application to multi-body gravitational regimes. In: The 63rd International Astronautical Congress, Naples, Italy (2012)

    Google Scholar 

  • Tallapragada, P.: Identifying dynamical boundaries and phase space transport using Lagrangian coherent structures. Ph.D. Dissertation, Virginia Tech. Univ. (2010)

  • Woodard, M., Folta, D., Woodfork, D.: ARTEMIS: the first mission to the lunar libration orbits. In: 21st International Symposium on Space Flight Dynamics, Toulouse, France (2009)

    Google Scholar 

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Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Rui Qi

  2. Department of Guidance, Navigation and Control, School of Astronautics, Beihang University, Beijing, 100191, China

    Shi Jie Xu

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  1. Rui Qi
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  2. Shi Jie Xu
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Correspondence to Rui Qi.

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Qi, R., Xu, S.J. Applications of lagrangian coherent structures to expression of invariant manifolds in astrodynamics. Astrophys Space Sci 351, 125–133 (2014). https://doi.org/10.1007/s10509-014-1819-9

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  • DOI: https://doi.org/10.1007/s10509-014-1819-9

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