[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Synchronization of Two Identical Restricted Planar Isosceles Three-Body-Problem and a Study on Possible Chaos Control

Discontinuity, Nonlinearity, and Complexity 2(2) (2013) 183--201 | DOI:10.5890/DNC.2013.04.007

Ayub Khan$^{1}$; Rimpi Pal$^{2}$

$^{1}$ Department of Mathematics, Jamia Millia Islamia University, New Delhi, India

$^{2}$ Department of Mathematics, University of Delhi, New Delhi, India

Download Full Text PDF

 

Abstract

In this paper, we have investigated Complete Synchronization, Anti- Synchronization and all possible cases of Hybrid Synchronization of two restricted planar isosceles three-body-problem evolving from different initial conditions using Active Control Technique. Further, the possible role of control functions in the control of Chaos is analyzed. Finally, numerical simulations are performed to illustrate the effectiveness of the proposed control techniques.

References

  1. [1]  Euler, L. (1765), De motu rectilineo trium corporum se mutuo attrahentium, Nov. Commun, Petrop, 11, 1-144.
  2. [2]  Lagrange, J.L. (1888), Mecanique Analylique, (Gauthier-Villars, Paris, 1888 ; Nauka, Moscow, 1952), 1
  3. [3]  Pecora, L.M. and Carroll, T.L. (1990), Synchronization in chaotic systems, Physical Review Letters, 64, 821-824.
  4. [4]  Kapitaniak, T. (1996), Controlling Chaos-Theoretical. Practical Methods in Non-linear Dynamics, Academic Pree, London.
  5. [5]  Chen, G. and Dong, X. (1998), From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore.
  6. [6]  Pikovsky, A.S., Rosenblum, M.G., and Kurths, J. (2001), Synchronization- A Unified approach to Non-linear Science, Cambridge University Press, Cambridge.
  7. [7]  Lakshmanan, M. andMurali, K. (1996), Chaos in Non-linear Oscillators: Controlling and Synchronization,World Scientific, Singapore.
  8. [8]  Fradkov, A.L. and Pogromsky, A.Y. (1996), Introduction to Control of Oscillations and Chaos, World Scientific, Singapore.
  9. [9]  Erjaee, G.H. (2009), Analytical justification of phase synchronization in chaotic systems, Chaos, Solitons & Fractals, The Interdisciplinary Journal of Nonlinear Science Nono and Quantum Technology, 39.
  10. [10]  Erjaee, G.H. and Momani, S. (2008), Phase synchronization in fractional differential chaotic, Physics Letters A, 372, 23502354.
  11. [11]  Bennett, M., Schatz, M.F., Rockwood, H., and Wiesenfeld (2002), Huygens' clock, Proceedings of the Royal Society London A, 458, 563-579.
  12. [12]  Cao, L.-Y. and Lai, Y-Ch. (1998), Antiphase synchronism in chaotic systems, Physical Review E, 58(1), 382-386.
  13. [13]  Liu, J., Ye, C., Zhang, S., and Song,W. (2000), Antiphase synchronization in coupled map lattices, Physics Letters A, 274, 27-29.
  14. [14]  Nakata, S., Mujata, T., Ojima, N., and Yoshikawa, K. (1998), Self-synchronization in coupled salt-water oscillators, Physica D, 115, 313-320.
  15. [15]  Uchida, A., Liu, Y., Fischer, I., Davis, P., and Aida, T. (2007), Chaotic antiphase dynamics and synchronization in multimode semiconductor lasers, Physical Review A, 64, (023801).
  16. [16]  Chuangdong Li, Qian Chen and Tingwen Huang (2008), Co-existence of anti-phase and complete synchronization in coupled Chen system via single variable, Chaos, Solitons and Fractals, 38, 461-464.
  17. [17]  Ott., Grebogi, C., and Yorke, J.A. (1990), Controlling chaos, Physical Review Letters, 64 1196-1199.
  18. [18]  Bai, E.W. and Lonngren,K.E. (1997) ,Synchronization of two Lorenz systems using active control, Chaos, Solitons and Fractals, 8, 51-58.
  19. [19]  Bai, E.W. and Lonngren, K.E. (2000), Sequential synchronization of two Lorenz systems using active control, Chaos, Solitons and Fractals, 11, 1041-1044.
  20. [20]  Ucar, A., Bai, E.W., and Lonngren,K.E. (2003), Synchronization of chaotic behavior in nonlinear Bloch equations, Physics Letters A, 314, 96-101.
  21. [21]  Tan, X., Zhang, J., and Yang, Y. (2003), Synchronizing chaotic systems using backstepping design, Chaos, Solitons and Fractals 16, 37-45.
  22. [22]  Yu, Y. and Zhang, S. (2004), Adaptive backstepping synchronization of uncertain chaotic systems, Chaos, Solitons and Fractals, 21, 643-649.
  23. [23]  Codreanu, S. (2003), Synchronization of spatiotemporal nonlinear dynamical systems by an active control, Chaos, Solitons and Fractals, 15, 507-510.
  24. [24]  Vincent, U.E. (2005), Synchronization of Rikitake chaotic attractor via active control, Physics Letters A, 343, 133-138.
  25. [25]  Ucar, A., Lonngren, K.E., and Bai, E.W. (2007), Chaos synchronization in RCL-shunted Josephson junction via active control, Chaos, Solitons and Fractals, 31, 105-111.
  26. [26]  Tang, F. and Wang, L. (2006), An adaptive active control for the modified Chuas circuit, Physics Letters A, 346, 342-346.
  27. [27]  Mahmoud, G.M., Aly, S.A., and Farghaly, A.A. (2007), On chaos synchronization of a complex two coupled dynamos system, Chaos, Solitons and Fractals, 33, 178-187.
  28. [28]  Vincent, U.E. (2008), Synchronization of identical and non-identical 4-D chaotic systems via active control, Chaos, Solitons and Fractals, 37, 1065-1075.
  29. [29]  Lei, Y., Xu,W., Shen, J., and Fang, T. (2006), Global synchronization of two parametrically excited systems using active control, Chaos, Solitons and Fractals, 28, 428-436.
  30. [30]  Lei, Y., Xu,W., and Xie,W. (2007), Synchronization of two chaotic four-dimensional systems using active control, Chaos, Solitons and Fractals, 32, 1823-1829.
  31. [31]  Kristic, M., Kanellakopoulo, I., and Kotkotovic (1995), Nonlinear and Adaptive Control Design, JohnWiley, New York.
  32. [32]  Harb, A.M. and Harb, B.A. (2004), Chaos control of third-order phase-locked loops using backstepping nonlinear controller, Chaos, Solitons and Fractals, 20, 719-723.
  33. [33]  Harb, A.M. (2004), Nonlinear chaos control in permanent magnet reluctance machine, Chaos, Solitons and Fractals, 19, 1217-1224.
  34. [34]  Yu, H., Feng, Z., and Wang, X. (2004), Nonlinear control for a class of hydraulic servo system, Journals of Zhejiang University-Science, 11, 1413-1417.
  35. [35]  Park, J.H. (2006), Synchronization of Genesio chaotic system via backstepping approach, Chaos, Solitons and Fractals, 27, 1369-1375.
  36. [36]  Ge, S.S., Wang, C., and Lee, T.H. (2000), Adaptive backstepping control of a class of chaotic systems, International Journal of Bifurcation and Chaos, 10, 1149-1156.
  37. [37]  Cors, Josep, M., Castilho, C., Vidal, C. (2009), The restricted planar isosceles three-body problem with nonnegative energy, Celestial Mechanics and Dynamical Astronomy, 103, 163-177.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates