Skip to main content
SpringerLink
Log in

The study of Newton–Raphson basins of convergence in the three-dipole problem

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We consider a system in which the charged particle orbits under the influence of the electromagnetic field of three dipoles located on a system of three celestial bodies. Using well-known bivariate iterative scheme, known as Newton–Raphson (NR) iterative scheme, we numerically evaluated the positions of the stationary points (SPs) or equilibrium points (EPs) or libration points (LPs) and the linked basins of convergence (BoCs), and we also evaluated their linear stability. Moreover, we unveiled how the parameters, entering the effective potential function, affect the convergence dynamics of the system. Moreover, we also unveiled how the involved parameters affect the geometry of the zero velocity curves (ZVCs). Further, the correlation with the required number of iterations and the regions of convergence as well as the probability distributions associated to the BoCs is illustrated. In order to quantify the degree of final-state uncertainty of the BoCs, the basin entropy (BE) and for the fractality of boundaries of BoCs, the boundary basin entropy (BBE) are computed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Alvarez-Ramírez, M., Barrabés, E.: Transport orbits in an equilateral restricted four-body problem. Celest. Mech. Dyn. Astr. 121, 191–210 (2015). https://doi.org/10.1007/s10569-014-9594-z

    Article  MathSciNet  Google Scholar 

  2. Abouelmagd, E.I., El-Shaboury, S.M.: Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies. Astrophys. Space Sci. 341, 331–341 (2012). https://doi.org/10.1007/s10509-012-1093-7

    Article  MATH  Google Scholar 

  3. Abouelmagd, E.I., Mostafa, A.: Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass. Astrophys. Space Sci. 357, 58 (2015). https://doi.org/10.1007/s10509-015-2294-7

    Article  Google Scholar 

  4. Abouelmagd, E.I., Mostafa, A., Guirao, J.L.G.: A first order automated lie transform. Int. J. Bifurc. Chaos 25(14), 1540026 (2015)

    Article  MathSciNet  Google Scholar 

  5. Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body problem. Int. J. Bifurc. Chaos 21, 2179–2193 (2011). https://doi.org/10.1142/S0218127411029707

    Article  MathSciNet  MATH  Google Scholar 

  6. Bosanac, N.: Exploring the influence of a three-body interaction added to the gravitational potential function in the circular restricted three-body problem: a numerical frequency analysis. M.Sc. Thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana (2012). https://engineering.purdue.edu/people/kathleen.howell.1/Publications/masters.html

  7. Bosanac, N., Howell, K.C., Fischbach, E.: Exploring the impact of a three-body interaction added to the gravitational potential function in the restricted three-body problem. In: Proceedings of the 23rd AAS/AIAA Space Flight Mechanics Meeting, February 2013, Hawaii (2013). AAS 13-490. https://engineering.purdue.edu/people/kathleen.howell.1/Publications/Conferences/2013_AAS_BosHowFis.pdf

  8. Daza, A., Wagemakers, A., Georgeot, B., Guéry-Odelin, D., Sanjuán, M.A.F.: Basin entropy: a new tool to analyze uncertainty in dynamical systems. Sci. Rep. 6, 31416 (2016). https://doi.org/10.1038/srep31416

    Article  Google Scholar 

  9. Daza, A., Georgeot, B., Guéry-Odelin, D., Wagemakers, A., Sanjuán, M.A.F.: Chaotic dynamics and fractal structures in experiments with cold atoms. Phys. Rev. A 95(1), 013629 (2017). https://doi.org/10.1103/PhysRevA.95.013629

    Article  MATH  Google Scholar 

  10. Desiniotis, C.D., Kazantzis, P.G.: The equilibrium configurations of the three-dipole problem. Astrophys. Space Sci. 202, 89 (1993). https://doi.org/10.1007/BF00626920

    Article  MATH  Google Scholar 

  11. Douskos, C.N.: Collinear equilibrium points of Hills problem with radiation and oblateness and their fractal basins of attraction. Astrophys. Space Sci. 326, 263–271 (2010). https://doi.org/10.1007/s10509-009-0213-5

    Article  MATH  Google Scholar 

  12. Goudas, C. L., Petsagourakis, E. G.: The three-dipole problem, Predietability, Stability and Chaos in N-body Dynamicals Systems, p. 355, (1991). https://www.springer.com/gp/book/9781468459999

  13. Goudas, C. L.: The N-dipole problem and the rings of Saturn, Predietability, Stability and Chaos in N-body Dynamicals Systems, p. 371 (1991). https://www.springer.com/gp/book/9781468459999

  14. Kalvouridis, T.J.: Parametric dependence of the collinear equilibria and their stability in the equatorial magnetic-binary problem. Astrophys. Space Sci. 150, 149–172 (1988)

    Article  Google Scholar 

  15. Kalvouridis, T.J.: Three-dimensional equilibria and their stability in the magnetic-binary problem. Astrophys. Space Sci. 159, 91–97 (1989)

    Article  Google Scholar 

  16. Kalvouridis, T.J.: The three-dimensional equilibria of magnetic-binary systems with oblate primaries. Celest. Mech. Dyn. Astron. 50(4), 301–311 (1991)

    Article  MathSciNet  Google Scholar 

  17. Kalvouridis, T.J.: Planar motions of a charged particle in a magnetic binary system with oblate primaries. Celest. Mech. Dyn. Astron. 58, 309–315 (1993)

    Article  Google Scholar 

  18. Kalvouridis, T., Mavraganis, A.: The symplectic character of orbits in the planar magnetic-binary problem. Acta Mech. 42, 135–141 (1982)

    Article  Google Scholar 

  19. Kalvouridis, T., Mavraganis, A.: The equatorial equilibrium configuration of the magnetic-binary problem. Celest. Mech. 35, 397–408 (1985)

    Article  Google Scholar 

  20. Kalvouridis, T.J., Mavraganis, A.G.: Symmetric motions in the equatorial magnetic-binary problem. Celest. Mech. 40, 177–196 (1987)

    Article  Google Scholar 

  21. Kalvouridis, T., Mavraganis, A., Pangalos, C.: The symplectic character of orbits in the three-dimensional magnetic-binary problem. Astrophys. Space Sci. 96, 255–259 (1983)

    Article  MathSciNet  Google Scholar 

  22. Kalvouridis, T., Mavraganis, A., Pangalos, C., Zagouras, C.: Areas of equatorial motion in the magnetic-binary problem. Celest. Mech. 37, 161–170 (1985)

    Article  Google Scholar 

  23. Mavraganis, A.: The magnetic-binaries problem: motion of a charged particle in the region of a magnetic-binary system. Astrophys. Space Sci. 54, 305–313 (1978)

    Article  Google Scholar 

  24. Markellos, V., Zagouras, C.: Effect of perturbation on the periodic solutions of the Stoermer problem. Astron. Astrophys. 61, 505–514 (1977)

    MATH  Google Scholar 

  25. Mittal, A., Agarwal, R., Suraj, M.S., Bisht, V.S.: Stability of LPs in the restricted four-body problem with variable mass. Astrophys. Space Sci. 361, 329 (2016). https://doi.org/10.1007/s10509-016-2901-2

    Article  Google Scholar 

  26. Mittal, A., Agarwal, R., Suraj, M.S., Arora, M.: On the photo-gravitational restricted four-body problem with variable mass. Astrophys. Space Sci. 363, 109 (2018). https://doi.org/10.1007/s10509-018-3321-2

    Article  MathSciNet  Google Scholar 

  27. Nagler, J.: Crash test for the Copenhagen problem. Phys. Rev. E 69, 066218 (2004). https://doi.org/10.1103/PhysRevE.69.066218

    Article  MathSciNet  Google Scholar 

  28. Nagler, J.: Crash test for the restricted three-body problem. Phys. Rev. E 71, 026227 (2005). https://doi.org/10.1103/PhysRevE.71.026227

    Article  MathSciNet  Google Scholar 

  29. Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013). https://doi.org/10.1007/s10509-012-1319-8

    Article  MATH  Google Scholar 

  30. Pathak, N., Thomas, V.O., Abouelmagd, E.I.: The perturbed photogravitational restricted three-body problem: analysis of resonant periodic orbits. Disc. Cont. Dyn. Syst. Ser. S 12(4–5), 849–875 (2019a)

    MathSciNet  MATH  Google Scholar 

  31. Pathak, N., Abouelmagd, E.I., Thomas, V.O.: On higher order resonant periodic orbits in the photo-gravitational planar restricted three-body problem with oblateness. J. Astronaut. Sci. 66(4), 475–505 (2019b). https://doi.org/10.1007/s40295-019-00178-z

    Article  Google Scholar 

  32. Robutel, P., Gabern, F.: The resonant structure of Jupiters Trojan asteroids: I. Long-term stability and diffusion. Month. Not. Roy. Astron. Soc. (2006). https://doi.org/10.1111/j.1365-2966.2006.11008.x

    Article  Google Scholar 

  33. Schwarz, R., Süli, Á., Dvorak, R., et al.: Stability of Trojan planets in multi-planetary systems. Celest. Mech. Dyn. Astr. 104, 69–84 (2009a). https://doi.org/10.1007/s10569-009-9210-9

    Article  MATH  Google Scholar 

  34. Schwarz, R., Süli, Á., Dvorak, R., et al.: Dynamics of possible Trojan planets in binary systems. Mon. Not. R. Astron. Soc. 398, 2085–2090 (2009b)

    Article  Google Scholar 

  35. Selim, H.H., Guirao, J.L.G., Abouelmagd, E.I.: Libration points in the restricted three-body problem: euler angles, existence and stability. Disc. Cont. Dyn. Syst. Ser. S 12(45), 703–710 (2019)

    MathSciNet  MATH  Google Scholar 

  36. Sharma, R.K., Rao, P.V.S.: Collinear equilibria and their characteristic exponents in the restricted three-body problem when the primaries are oblate spheroids. Celest. Mech. 12, 189–201 (1975). https://doi.org/10.1007/BF01230211

    Article  MATH  Google Scholar 

  37. Störmer, C.: On the trajectories of electric corpuscles in space under the influence of terrestrial magnetism applied to the aurora borealis and to magnetic disturbances. Arch. Math. Naturvidens. Christiania XXVIII(2) (1906)

  38. Suraj, M.S., Hassan, M.R., Asique, M.C.: The photo-gravitational R3BP when the primaries are heterogeneous spheroid with three layers. J. Astronaut. Sci. 61, 133–155 (2014). https://doi.org/10.1007/s40295-014-0026-9

    Article  Google Scholar 

  39. Suraj, M.S., Aggarwal, R., Arora, M.: On the restricted four-body problemwith the effect of small perturbations in the Coriolis and centrifugal forces. Astrophys. Space Sci. 362, 159 (2017). https://doi.org/10.1007/s10509-017-3123-y

    Article  Google Scholar 

  40. Suraj, M.S., Aggarwal, R., Kumari, S., Asique, M.C.: Out-of-plane equilibrium points and regions of motion in the photogravitational R3BP when the primaries are hetrogeneous spheroid with three layers. New Astron. 63, 15–26 (2018a). https://doi.org/10.1016/j.newast.2018.02.005

    Article  Google Scholar 

  41. Suraj, M.S., Mittal, A., Arora, M., et al.: Exploring the fractal basins of convergence in the restricted four-body problem with oblateness. Int. J. Non-Linear Mech. 102, 62–71 (2018b)

    Article  Google Scholar 

  42. Suraj, M.S., Abouelmagd, E.I., Aggarwal, R., Mittal, A.: The analysis of restricted five-body problem within frame of variable mass. New Astron. 70, 12–21 (2019). https://doi.org/10.1016/j.newast.2019.01.002

    Article  Google Scholar 

  43. Suraj, M.S., Sachan, P., Aggarwal, R., Mittal, A.: On the fractal basins of convergence of the LPs in the axisymmetric five-body problem: the convex configuration. Int. J. Nonlinear Mech. 109, 80–106 (2019). https://doi.org/10.1016/j.ijnonlinmec.2018.11.005

    Article  Google Scholar 

  44. Suraj, M.S., Sachan, P., Zotos, E.E., Mittal, A., Aggarwal, R.: On the Newton–Raphson basins of convergence associated with the LPs in the axisymmetric restricted five-body problem: the concave configuration. Int. J. Nonlinear Mech. 112, 25–47 (2019). https://doi.org/10.1016/j.ijnonlinmec.2019.02.013

    Article  Google Scholar 

  45. Suraj, M.S., Sachan, P., Aggarwal, R., Mittal, A.: The effect of small perturbations in the Coriolis and centrifugal forces in the axisymmetric restricted five-body problem. Astrophys. Space Sci. 364, 44 (2019). https://doi.org/10.1007/s10509-019-3528-x

    Article  MathSciNet  Google Scholar 

  46. Suraj, M.S., Aggarwal, R., Asique, M.D., Mittal, A., Sachan, P.: On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence. Astrophys. Space Sci. 364, 87 (2019). https://doi.org/10.1007/s10509-019-3575-3

    Article  MathSciNet  Google Scholar 

  47. Suraj, M.S., Aggarwal, R., Asique, M.C., Mittal, A.: On the modified circular restricted three-body problem with variable mass. New Astronomy, 84, 101510 (2021). https://www.sciencedirect.com/science/article/pii/S0020746218306942

  48. Szebehley, V.: Stability of the points of equilibrium in the restricted problem. Astron. J. 72, 7 (1967)

    Article  Google Scholar 

  49. Zotos, E.E.: Fractal basins of attraction in the planar circular restricted three-body problem with oblateness and radiation pressure. Astrophys. Space Sci. 361, 181 (2016). https://doi.org/10.1007/s10509-016-2769-1

  50. Zotos, E.E.: Revealing the basins of convergence in the planar equilateral restricted four-body problem. Astrophys. Space Sci. 362, 2 (2017a). https://doi.org/10.1007/s10509-016-2973-z

    Article  MathSciNet  Google Scholar 

  51. Zotos, E.E.: Equilibrium points and basins of convergence in the linear restricted four-body problem with angular velocity. Chaos Solitons Fract. 101, 8–19 (2017b). https://doi.org/10.1016/j.chaos.2017.05.003

    Article  MathSciNet  MATH  Google Scholar 

  52. Zotos, E.E.: Basins of convergence of equilibrium points in the pseudo-Newtonian planar circular restricted three-body problem. Astrophys. Space Sci. 362, 195 (2017c). https://doi.org/10.1007/s10509-017-3172-2

    Article  MathSciNet  Google Scholar 

  53. Zotos, E.E.: Comparing the fractal basins of attraction in the Hill problem with oblateness and radiation. Astrophys. Space Sci. 362, 190 (2017d). https://doi.org/10.1007/s10509-017-3169-x

    Article  MathSciNet  Google Scholar 

  54. Zotos, E.E., Suraj, M.S.: Basins of attraction of equilibrium points in the planar circular restricted five-body problem. Astrophys. Space Sci. 363, 20 (2018a). https://doi.org/10.1007/s10509-017-3240-7

    Article  MathSciNet  Google Scholar 

  55. Zotos, E.E.: On the Newton–Raphson basins of convergence of the out-of-plane equilibrium points in the Copenhagen problem with oblate primaries. Int. J. Non-Linear Mech. 103, 93–103 (2018b). https://doi.org/10.1016/j.ijnonlinmec.2018.05.002

  56. Zotos, E.E., Suraj, M.S., Aggarwal, R., et al.: On the convergence dynamics of the sitnikov problem with non-spherical primaries. Int. J. Appl. Comput. Math. 5, 43 (2019). https://doi.org/10.1007/s40819-019-0627-xhttps://doi.org/10.1016/j.ijnonlinmec.2018.05.002

Download references

Funding

Dr. Rajiv Aggarwal was partially supported by Department of Science and Technology, India, under scheme MATRICS (MTR/2018/000442).

Author information

Authors and Affiliations

  1. Department of Mathematics, Sri Aurobindo College, University of Delhi, New Delhi, Delhi, 110017, India

    Md Sanam Suraj

  2. Center for Fundamental Research in Space Dynamics and Celestial Mechanics (CFRSC), New Delhi, Delhi, India

    Md Sanam Suraj, Rajiv Aggarwal & Kumari Shalini

  3. Department of Mathematics, Deshbandhu College, University of Delhi, New Delhi, Delhi, 110019, India

    Rajiv Aggarwal

  4. DBC i4 Centre, Deshbandhu College, University of Delhi, New Delhi, Delhi, 110019, India

    Rajiv Aggarwal & Md Chand Asique

  5. Department of Physics, Deshbandhu College, University of Delhi, New Delhi, Delhi, 110019, India

    Md Chand Asique

  6. Department of Mathematics, Zakir Hussain College, University of Delhi, New Delhi, Delhi, 110002, India

    Kumari Shalini

Authors
  1. Md Sanam Suraj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajiv Aggarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Md Chand Asique
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kumari Shalini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Md Sanam Suraj.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Data availability statements

All data generated or analyzed during this study are included in this published article (and its supplementary information files).

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suraj, M.S., Aggarwal, R., Asique, M.C. et al. The study of Newton–Raphson basins of convergence in the three-dipole problem. Nonlinear Dyn 107, 829–854 (2022). https://doi.org/10.1007/s11071-021-07029-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07029-3

Keywords

Search

Navigation