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Enhanced Visualization and Autonomous Extraction of Poincaré Map Topology

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Abstract

Poincaré maps supply vital descriptions of dynamical behavior in spacecraft trajectory analysis, but the puncture plot, the standard display method for maps, typically requires significant external effort to extract topology. This investigation presents adaptations of topology-based methods to compute map structures in multi-body dynamical environments. In particular, a scalar field visualization technique enhances the contrast between quasi-periodic and chaotic regimes. Also, an autonomous method is outlined to extract map topology in the planar circular restricted three-body problem. The resulting topological skeleton supplies a network of design options through the interconnectivity of orbital structures.

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References

  1. Anderson, R.L., Lo, M.W.: A dynamical systems analysis of resonant flybys: Ballistic case. J. Astronaut. Sci. 58(2), 167–194 (2011)

    Article  Google Scholar 

  2. Cabral, B., Leedom, L.C.: Imaging Vector Fields Using Line Integral Convolution, pp 263–270. ACM, New York, NY, USA (1993)

    Google Scholar 

  3. Danby, J.M.: Fundamentals of Celestial Mechanics, 2nd edn. Willmann-Bell, Inc., Richmond, Virginia (1992)

    Google Scholar 

  4. Demeyer, J., Gurfil, P.: Transfer to distant retrograde orbits using manifold theory. J. Guid. Control Dyn. 30, 5 (2007)

    Article  Google Scholar 

  5. Doedel, E.J., Romanov, V.A., Paffenroth, R.C., Keller, H.B., Dichmann, D.J., Galan-Vioque, J., Vanderbauwhede, A.: Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem. Int. J. Bifurcation Chaos 17(8), 2625–2677 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. England, J., Krauskopf, B., Osinga, H.: Computing one-dimensional global manifolds of Poincaré maps by continuation. SIAM J. Appl. Dyn. Syst. 4(4), 1008–1041 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gómez, G., Koon, W.S., Lo, M.W., Marsden, J.E., Masdemont, J., Ross, S.D.: Connecting orbits and invariant manifolds in the spatial restricted three-body problem. Nonlinearity 17, 1571–1606 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York. New York (1983)

  9. Haapala, A., Howell, K.C.: Trajectory design strategies applied to temporary comet capture including Poincaré maps and invariant manifolds. Celest. Mech. Dyn. Astron. 116(3), 299–323 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics, 2nd edn. Springer-Verlag, New York, New York (1992)

    Book  MATH  Google Scholar 

  12. Lo, M.W., Anderson, R.L., Lam, T., Whiffen, G.: The role of invariant manifolds in low thrust trajectory design (Part III). In: AAS/AIAA Spaceflight Dynamics Conference. Tampa, Florida. Paper AAS 06-190 (2006)

  13. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007)

    Google Scholar 

  14. Sanderson, A.R., Chen, G., Tricoche, X., Pugmire, D., Kruger, S., Breslau, J.: Analysis of recurrent patterns in toroidal magnetic fields. In: Visualization Information Visualization 2010, IEEE Transactions on Visualization and Computer Graphics, vol. 16 (2010)

  15. Schlei, W.: An Application of Visual Analytics to Spacecraft Trajectory Design. M.S. Thesis, School of Aeronautics and Astronautics, Purdue University. West Lafayette, Indiana (2011)

    Google Scholar 

  16. Shoemake, K.: Rational approximation. In: Paeth, A.W. (ed.) Graphics Gems V, vol. 25–32. Academic Press, San Diego (1995)

    Google Scholar 

  17. Strogatz, S.H.: Nonlinear Dynamics and Chaos. Westview Press, Perseus Books Publishing, Cambridge (1994)

    Google Scholar 

  18. Thrill, M.: A more precise rounding algorithm for rational numbers. Computing 82(2–3), 189–198 (2008)

    Article  MathSciNet  Google Scholar 

  19. Tricoche, X., Garth, C., Sanderson, A.: Visualization of topological structures in area-preserving maps. IEEE Trans. Vis. Comput. Graph. 17(12), 1765–1774 (2011)

    Article  Google Scholar 

  20. Vaquero, M., Howell, K.C.: Leveraging resonant orbit manifolds to design transfers between libration point orbits in multi-body regimes. In: 23rd AAS/AIAA Space Flight Mechanics Meeting Kauai Hawaii (2013)

  21. Vaquero, M., Howell, K.C.: Transfer design exploiting resonant orbits and manifolds in the saturn-titan system. J. Spacecr. Rocket. 50(5), 1069–1085 (2013)

    Article  Google Scholar 

  22. Villac, B.F., Scheeres, D.J.: Escaping trajectories in the hill three-body problem and applications. J.Guid. Control, Dyn. 26(2), 224–232 (2003)

    Article  Google Scholar 

  23. Yip, K.M.K.: KAM: A System for Intelligently Guiding Numerical Experimentation by Computer. The MIT Press (1991)

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Acknowledgments

The authors are grateful to Rune and Barbara Eliasen for their support in funding the Rune and Barbara Eliasen Visualization Laboratory at Purdue University. Also, the authors wish to acknowledge Visualization Sciences Group (the developers of Avizo®) for programming and implementation assistance with the visualization tools employed in this work. A significant portion of this research is supported as part of the NSF CAREER Program Award #1150000: Efficient Structural Analysis of Multivariate Fields for Scalable Visualizations. This effort is also supported by the Computer Science Department and the School of Aeronautics and Astronautics at Purdue University.

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Authors and Affiliations

  1. Armstrong Hall, Purdue University, 701 West Stadium Avenue, West Lafayette, IN, 47907-2045, USA

    Wayne Schlei & Kathleen C. Howell

  2. Lawson Hall, Purdue University, 305 N University Street, West Lafayette, IN, 47907-2045, USA

    Xavier Tricoche

  3. University of Kaiserslautern, PO Box 3049, Kaiserslautern, 67653, Germany

    Christoph Garth

Authors
  1. Wayne Schlei
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  2. Kathleen C. Howell
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  3. Xavier Tricoche
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  4. Christoph Garth
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Corresponding author

Correspondence to Wayne Schlei.

Additional information

A previous version of this paper received the Best Paper Award for the AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, August 2013.

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Schlei, W., Howell, K.C., Tricoche, X. et al. Enhanced Visualization and Autonomous Extraction of Poincaré Map Topology. J of Astronaut Sci 61, 170–197 (2014). https://doi.org/10.1007/s40295-015-0042-4

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  • DOI: https://doi.org/10.1007/s40295-015-0042-4

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