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Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies

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Abstract

This paper studies the distribution of characteristic multipliers, the structure of submanifolds, the phase diagram, bifurcations, and chaotic motions in the potential field of rotating highly irregular-shaped celestial bodies (hereafter called irregular bodies). The topological structure of the submanifolds for the orbits in the potential field of an irregular body is shown to be classified into 34 different cases, including six ordinary cases, three collisional cases, three degenerate real saddle cases, seven periodic cases, seven period-doubling cases, one periodic and collisional case, one periodic and degenerate real saddle case, one period-doubling and collisional case, one period-doubling and degenerate real saddle case, and four periodic and period-doubling cases. The different distribution of the characteristic multipliers has been shown to fix the structure of the submanifolds, the type of orbits, the dynamical behaviour and the phase diagram of the motion. Classifications and properties for each case are presented. Moreover, tangent bifurcations, period-doubling bifurcations, Neimark–Sacker bifurcations, and the real saddle bifurcations of periodic orbits in the potential field of an irregular body are discovered. Submanifolds appear to be Mobius strips and Klein bottles when the period-doubling bifurcation occurs.

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Acknowledgments

This research was supported by the National Basic Research Program of China (973 Program, 2012CB720000), the State Key Laboratory Foundation of Astronautic Dynamics (No. 2014ADL0202), and the National Natural Science Foundation of China (No. 11372150).

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Authors and Affiliations

  1. State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an, 710043, China

    Yu Jiang

  2. School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Yu Jiang, Yang Yu & Hexi Baoyin

Authors
  1. Yu Jiang
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  2. Yang Yu
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  3. Hexi Baoyin
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Corresponding author

Correspondence to Yu Jiang.

Appendix

Appendix

See Table 1.

Table 1 Classifications and properties for the (a) ordinary cases, (b) periodic cases, (c) period-doubling cases, (d) collisional cases, (e) degenerate real saddle cases, (f) mixed cases
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Jiang, Y., Yu, Y. & Baoyin, H. Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies. Nonlinear Dyn 81, 119–140 (2015). https://doi.org/10.1007/s11071-015-1977-5

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  • DOI: https://doi.org/10.1007/s11071-015-1977-5

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