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Dynamical Properties of Body with Variable Mass in a Fifth-degree Hénon–Heiles System

Abstract

The aim of this work is to investigate the motion properties of the infinitesimal body in the frame of the fifth-degree generalized Hénon–Heiles (GHH) potential. We determine the equations of motion of the infinitesimal body, where its mass varies according to the Jeans law, within the framework of the GHH potential. We illustrate numerically the positions of stationary points and their stability, and we found that all these points are unstable. Further, the dynamical features such as Poincaré surfaces of section, the trajectory allocations, the basins of attraction, the permitted and prohibited regions of motion are studied for each stationary point.

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REFERENCES

  1. M. Hénon and C. Heiles, Astron. J. 69, 73 (1964).

    ADS  Article  Google Scholar 

  2. F. Verhulst, Phil. Trans. R. Soc. London, Ser. A 290 (1375), 435 (1979).

    ADS  Article  Google Scholar 

  3. T. De Zeeuw, Mon. Not. R. Astron. Soc. 215, 731 (1984).

    ADS  Article  Google Scholar 

  4. J. Aguirre, J. C. Vallejo, and M. A. F. Sanjuán, Phys. Rev. E 64, 066208 (2001).

    ADS  Article  Google Scholar 

  5. A. A. Ansari and E. I. Abouelmagd, Mod. Phys. Lett. A 36, 2150150 (2021).

    ADS  Article  Google Scholar 

  6. F. L. Dubeibe, A. Riaño Doncel, and E. E. Zotos, Phys. Lett. A 382, 904 (2017).

    ADS  Article  Google Scholar 

  7. E. E. Zotos, A. Riaño Doncel, and F. L. Dubeibe, Int. J. Non-Lin. Mech. 99, 218 (2018).

    Google Scholar 

  8. E. E. Zotos, W. Chen, E. I. Abouelmagd, and H. Han, Chaos, Solitons Fractals 134, 109704 (2020).

    MathSciNet  Article  Google Scholar 

  9. H. H. Selim, J. L. Guirao, and E. I. Abouelmagd, Discrete Contin. Dyn. Syst. S 12, 703 (2019).

    Article  Google Scholar 

  10. E. I. Abouelmagd and A. A. Ansari, New Astron. 73, 101282 (2019).

    Article  Google Scholar 

  11. N. Pathak, E. I. Abouelmagd, and V. Thomas, J. Astronaut. Sci. 66, 475 (2019).

    ADS  Article  Google Scholar 

  12. N. Pathak, V. Thomas, and E. I. Abouelmagd, Discrete Contin. Dyn. Syst. S 12, 849 (2019).

    Article  Google Scholar 

  13. J. Singh and B. Ishwar, Celest. Mech. 35, 201 (1985).

    ADS  Article  Google Scholar 

  14. L. G. Lukyanov, Astron. Lett. 35, 349 (2009).

    ADS  Article  Google Scholar 

  15. M. J. Zhang, C. Y. Zhao, and Y. Q. Xiong, Astrophys. Space Sci. 337, 107 (2012).

    ADS  Article  Google Scholar 

  16. J. H. Jeans, Astronomy and Cosmogony (Cambridge Univ. Press, Cambridge, 1928).

    MATH  Google Scholar 

  17. I. V. Meshcherskii, Works on the Mechanics of Bodies of Variable Mass (GITTL, Moscow, 1952) [in Russian].

    MATH  Google Scholar 

  18. A. S. Beevi and R. K. Sharma, Astrophys. Space Sci. 340, 245 (2012).

    ADS  Article  Google Scholar 

  19. E. I. Abouelmagd and S. M. El-Shaboury, Astrophys. Space Sci. 341, 331 (2012).

    ADS  Article  Google Scholar 

  20. E. I. Abouelmagd and A. Mostafa, Astrophys. Space Sci. 357, 1 (2015).

    Article  Google Scholar 

  21. E. I. Abouelmagd, A. A. Ansari, M. S. Ullah, and J. L. G. Guirao, Astron. J. 160, 216 (2020).

    ADS  Article  Google Scholar 

  22. A. A. Ansari, Ital. J. Pure Appl. Math. 38, 581 (2017).

    Google Scholar 

  23. A. A. Ansari, Z. A. Alhusain, and K. Rabah, Punjab Univ. J. Math. 51 (5), 107 (2019).

    Google Scholar 

  24. J. Nagler, Phys. Rev. E 69, 066218 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  25. J. Nagler, Phys. Rev. E 71, 026227 (2005).

    ADS  MathSciNet  Article  Google Scholar 

  26. J. Aguirre, R. L. Viana, and M. A. F. Sanjuan, Rev. Mod. Phys. 81, 333 (2009).

    ADS  Article  Google Scholar 

  27. A. Daza, A. Wagemakers, B. Georgeot, D. Guéry-Odelin, and M. A. F. Sanjuan, Sci. Rep. 6, 31416 (2016).

    ADS  Article  Google Scholar 

  28. A. A. Ansari and S. N. Prasad, Astron. Lett. 46, 275 (2020).

    ADS  Article  Google Scholar 

  29. A. A. Ansari, Sultan Qaboos Univ. J. Sci. 25, 61 (2020).

    Google Scholar 

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ACKNOWLEDGMENTS

Authors are thankful to the anonymous referee for the valuable comments by which we succeeded to improve our manuscript at this end.

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Correspondence to Elbaz I. Abouelmagd or Abdullah A. Ansari.

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Abouelmagd, E.I., Ansari, A.A. Dynamical Properties of Body with Variable Mass in a Fifth-degree Hénon–Heiles System. Astron. Rep. 66, 64–74 (2022). https://doi.org/10.1134/S1063772922010012

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  • DOI: https://doi.org/10.1134/S1063772922010012

Keywords:

  • fifth-order
  • mass variation effect
  • generalized Hénon–Heiles model
  • Poincaré surfaces of section
  • trajectories