Abstract
This paper presents a method for measuring the periodicity of quasi-periodic trajectories by applying discrete Fourier transform (DFT) to the trajectories and analyzing the frequency domain within the concept of entropy. Having introduced the concept of entropy, analytical derivation and numerical results indicate that entropies increase as a logarithmic function of time. Periodic trajectories typically have higher entropies, and trajectories with higher entropies mean the periodicities of the motions are stronger. Theoretical differences between two trajectories expressed as summations of trigonometric functions are also derived analytically. Trajectories in the Henon-Heiles system and the circular restricted three-body problem (CRTBP) are analyzed with the indicator entropy and compared with orthogonal fast Lyapunov indicator (OFLI). The results show that entropy is a better tool for discriminating periodicity in quasiperiodic trajectories than OFLI and can detect periodicity while excluding the spirals that are judged as periodic cases by OFLI. Finally, trajectories in the vicinity of 243 Ida and 6489 Golevka are considered as examples, and the numerical results verify these conclusions. Some trajectories near asteroids look irregular, but their higher entropy values as analyzed by this method serve as evidence of frequency regularity in three directions. Moreover, these results indicate that applying DFT to the trajectories in the vicinity of irregular small bodies and calculating their entropy in the frequency domain provides a useful quantitative analysis method for evaluating orderliness in the periodicity of quasi-periodic trajectories within a given time interval.
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References
M. J. S. Belton, J. Veverka, P. Thomas, P. Helfenstein, D. Simonelli, C. Chapman, M. E. Davies, R. Greeley, R. Greenberg, J. Head, S. Murchie, K. Klaasen, T. V. Johnson, A. McEwen, D. Morrison, G. Neukum, F. Fanale, C. Anger, M. Carr, and C. Pilcher, Science 257, 1647 (1992).
D. S. Lauretta, and OSIRIS-Rex Team, in An Overview of the OSIRISREx Asteroid Sample Return Mission: Proceedings of the 43rd Lunar and Planetary Science Conference, The Woodlands, Texas (2012).
J. Veverka, B. Farquhar, M. Robinson, P. Thomas, S. Murchie, A. Harch, P. G. Antreasian, S. R. Chesley, J. K. Miller, W. M. Owen, B. G. Williams, D. Yeomans, D. Dunham, G. Heyler, M. Holdridge, R. L. Nelson, K. E. Whittenburg, J. C. Ray, B. Carcich, A. Cheng, C. Chapman, J. F. Bell, M. Bell, B. Bussey, B. Clark, D. Domingue, M. J. Gaffey, E. Hawkins, N. Izenberg, J. Joseph, R. Kirk, P. Lucey, M. Malin, L. McFadden, W. J. Merline, C. Peterson, L. Prockter, J. Warren, and D. Wellnitz, Nature 413, 390 (2001).
K. H. Glassmeier, H. Boehnhardt, D. Koschny, E. Kührt, and I. Richter, Space Sci. Rev. 128, 1 (2007).
A. Tsuchiyama, M. Uesugi, T. Matsushima, T. Michikami, T. Kadono, T. Nakamura, K. Uesugi, T. Nakano, S. A. Sandford, R. Noguchi, T. Matsumoto, J. Matsuno, T. Nagano, Y. Imai, A. Takeuchi, Y. Suzuki, T. Ogami, J. Katagiri, M. Ebihara, T. R. Ireland, F. Kitajima, K. Nagao, H. Naraoka, T. Noguchi, R. Okazaki, H. Yurimoto, M. E. Zolensky, T. Mukai, M. Abe, T. Yada, A. Fujimura, M. Yoshikawa, and J. Kawaguchi, Science 333, 1125 (2011).
C. T. Russell, and C. A. Raymond, Space. Sci. Rev. 163, 3 (2011).
Y. Guo, and R. W. Farquhar, Space. Sci. Rev. 140, 49 (2008).
X. Wang, Y. Jiang, and S. Gong, Astrophys. Space Sci. 353, 105 (2014), arXiv: 1403.5025
X. Liu, H. Baoyin, and X. Ma, Astrophys. Space Sci. 333, 409 (2011), arXiv: 1108.4636
Y. Jiang, J. A. Schmidt, H. Li, X. Liu, and Y. Yang, Astrodynamics 2, 69 (2018).
X. Zeng, and K. T. Alfriend, Astrodynamics 1, 41 (2017).
L. Lan, Y. Ni, Y. Jiang, and J. Li, Acta Mech. Sin. 34, 214 (2018).
H. Yang, H. Baoyin, X. Bai, and J. Li, Astrophys. Space Sci. 362, 27 (2017).
Y. Jiang, H. Baoyin, X. Wang, Y. Yu, H. Li, C. Peng, and Z. Zhang, Nonlinear Dyn. 83, 231 (2016).
A. Riaguas, A. Elipe, and M. Lara, Celestial Mech. Dynamical Astron. 73, 169 (1999).
S. Gutierrez-Romero, J. F. Palacian, and P. Yanguas, Monogr Real Acad Cienc Zaragoza 25, 137 (2004).
W. Hu, and D. J. Scheeres, Planet. Space Sci. 52, 685 (2004).
W. D. Hu, and D. J. Scheeres, Chin. J. Astron. Astrophys. 8, 108 (2008).
X. D. Liu, H. X. Baoyin, and X. R. Ma, Sci. China-Phys. Mech. Astron. 56, 818 (2013).
X. Zeng, F. Jiang, J. Li, and H. Baoyin, Astrophys. Space Sci. 356, 29 (2015).
D. J. Scheeres, S. J. Ostro, R. S. Hudson, and R. A. Werner, Icarus 121, 67 (1996).
R. A. Werner, Celestial Mech. Dynamical Astron. 59, 253 (1994).
X. Zeng, K. T. Alfriend, and S. R. Vadali, J. Guid. Control Dyn. 37, 674 (2014).
X. Zeng, and X. Liu, IEEE Trans. Aerosp. Electron. Syst. 53, 1221 (2017).
Y. Yu, and H. Baoyin, Mon. Not. R. Astron. Soc. 427, 872 (2012).
K. Meyer, G. Hall, and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, 2nd ed. (Springer-Verlag, New York, 2009), Chapt. 9.
Y. Ni, H. Baoyin, and J. Li, Orbit Dynamics in the Vicinity of Asteroids with Solar Perturbation: Proceedings of the International Astronautical Congress, Vol. 7, (2014), pp. 4610–4620.
Y. Jiang, Y. Yu, and H. Baoyin, Nonlinear Dyn. 81, 119 (2015).
Y. Jiang, H. Baoyin, and H. Li, Astrophys. Space Sci. 360, 63 (2015), arXiv: 1511.07926
J. E. Marsden, and T. S. Ratiu, Introduction to Mechanics and Symmetry (Springer-Verlag, New York, 1999), Chapt. 5.
R. A. Broucke, and A. Elipe, Reg. Chaot. Dyn. 10, 129 (2005).
Y. Yu, H. Baoyin, and Y. Jiang, Mon. Not. R. Astron. Soc. 453, 3270 (2015).
Y. Ni, Y. Jiang, and H. Baoyin, Astrophys. Space Sci. 361, 170 (2016), arXiv: 1604.07226
D. C. Davis, and K. C. Howell, Acta Astronaut. 69, 1038 (2011).
D. C. Davis, and K. C. Howell, J. Guid. Control Dyn. 35, 116 (2012).
D. J. Scheeres, Acta Astronaut. 72, 1 (2012).
D. J. Scheeres, J. Guid. Control Dyn. 35, 987 (2012).
T. G. G. Chanut, O. C. Winter, and M. Tsuchida, Mon. Not. R. Astron. Soc. 438, 2672 (2014).
A. Elipe, and M. Lara, J Astronaut. Sci. 51, 391 (2003).
E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University Press, New York, 2002), Chapt. 6.
H. L. Swinney, and J. P. Gollub, Phys. Today 31, 41 (1978).
P. Robutel, and J. Laskar, Icarus 152, 4 (2001).
D. A. Dei Tos, and F. Topputo, Adv. Space Res. 59, 2117 (2017).
G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, Meccanica 15, 9 (1980).
M. Fouchard, E. Lega, C. Froeschlé, and C. Froeschlé, Celest. Mech. Dyn. Astron. 83, 205 (2002).
P. M. Cincotta, and C. Simó, Astron. Astrophys. Suppl. Ser. 147, 205 (2000).
A. N. Kolmogorov, Proc. USSR Acad. Sci. 119, 861 (1958).
Y. G. Sinai, Proc. USSR Acad. Sci. 124, 768 (1959).
M. Henon, and C. Heiles, Astron. J. 69, 73 (1964).
V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies (Academic, New York, 1967), p. 126.
X. Zeng, S. Gong, J. Li, and K. T. Alfriend, J. Guidance Control Dyn. 39, 1223 (2016).
H. W. Yang, X. Y. Zeng, and H. Baoyin, Res. Astron. Astrophys. 15, 1571 (2015).
M. Hénon, Ann. Astrophys. 28, 499 (1965).
L. Wilson, K. Keil, and S. J. Love, Meteoritics Planet. Sci. 34, 479 (1999).
D. Vokrouhlický, D. Nesvorný, and W. F. Bottke, Nature 425, 147 (2003).
S. R. Chesley, S. J. Ostro, D. Vokrouhlicky, D. Capek, J. D. Giorgini, M. C. Nolan, J. L. Margot, A. A. Hine, L. A. M. Benner, and A. B. Chamberlin, Science 302, 1739 (2003).
R. S. Hudson, S. J. Ostro, R. F. Jurgens, K. D. Rosema, J. D. Giorgini, R. Winkler, R. Rose, D. Choate, R. A. Cormier, C. R. Franck, R. Frye, D. Howard, D. Kelley, R. Littlefair, M. A. Slade, L. A. M. Benner, M. L. Thomas, D. L. Mitchell, P. W. Chodas, D. K. Yeomans, D. J. Scheeres, P. Palmer, A. Zaitsev, Y. Koyama, A. Nakamura, A. W. Harris, and M. N. Meshkov, Icarus 148, 37 (2000).
C. E. Neese, Small Body Radar Shape Models V2.0. EAR-A-5-DDRRADARSHAPE-MODELS-V2.0 (NASA Planetary Data System, 2004).
Y. Yu, and H. Baoyin, Astron. J. 143, 62 (2012).
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Citation: Y. Ni, K. Turitsyn, H. Baoyin, and J. Li, Entropy method of measuring and evaluating periodicity of quasi-periodic trajectories, Sci. China-Phys. Mech. Astron. 61, 064511 (2018), https://doi.org/10.1007/s11433-017-9161-8
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Ni, Y., Turitsyn, K., Baoyin, H. et al. Entropy method of measuring and evaluating periodicity of quasi-periodic trajectories. Sci. China Phys. Mech. Astron. 61, 064511 (2018). https://doi.org/10.1007/s11433-017-9161-8
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DOI: https://doi.org/10.1007/s11433-017-9161-8