Skip to main content
SpringerLink
Log in

Combination–combination synchronisation of time-delay chaotic systems for unknown parameters with uncertainties and external disturbances

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

In this article, an adaptive control method is proposed to study the combination–combination synchronisation phenomenon of four non-identical time-delayed chaotic systems for fully unknown parameters with parametric uncertainties and external disturbances. Based on the Lyapunov–Krasovskii functional theory, an appropriate adaptive controller is constructed so that a globally and asymptotically stable synchronisation state can be established between the master and the slave systems. Unknown parameters are identified by designing suitable parameter update laws. To elaborate the presented scheme, double-delay Rossler and time-delay Chen systems are considered as the master systems and time-delay Shimizu–Morioka and time-delay modified Lorenz systems are considered as the slave systems. Numerical simulations are presented to justify the theoretical analysis.

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

Similar content being viewed by others

References

  1. L M Pecora and T L Carroll, Phys. Rev. Lett. 64, 821 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  2. M Srivastava, S P Ansari, S K Agrawal, S Das and A Y T Leung, Nonlinear Dyn. 76, 905 (2014)

    Article  Google Scholar 

  3. M M Al-Sawalha and M S M Noorani, Phys. Lett. A 373, 2852 (2009)

    Article  ADS  Google Scholar 

  4. W Jawaadaa, M S M Noorani and M Mossa Al-sawalha, Nonlinear Anal. Real World Appl. 13, 2403 (2012)

    Article  MathSciNet  Google Scholar 

  5. Z Cheng and J Cao, Appl. Math. Comput. 218, 2122 (2011)

    MathSciNet  Google Scholar 

  6. G Feng and J Cao, Adv. Differ. Equ. 2013, 24 (2013)

    Article  Google Scholar 

  7. X Liao and G Chen, Int. J. Bifurc. Chaos 13, 207 (2003)

    Article  Google Scholar 

  8. Z Hong-Yan, M Le-Quan, Z Geng and C Guan-Rong, Chin. Phys. Lett. 30, 040502 (2013)

    Article  Google Scholar 

  9. W He and J Cao, Chaos 19, 013118 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  10. Y Xu, W Zhou, J Fang and W Sun, Phys. Lett. A 374, 3441 (2010)

    Article  ADS  Google Scholar 

  11. S Wang, H Yao, S Zheng and Y Xie, Commun. Nonlinear Sci. Numer. Simul. 17, 2997 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  12. H Du, Q Zeng, C Wang and M Ling, Nonlinear Anal. Real World Appl. 11, 705 (2010)

    Article  MathSciNet  Google Scholar 

  13. L Runzi, W Yinglan and D Shucheng, Chaos 21, 043114 (2011)

    Article  ADS  Google Scholar 

  14. J Sun, Y Shen, G Zhang, C Xu and G Cui, Nonlinear Dyn. 73, 1211 (2013)

    Article  Google Scholar 

  15. J Sun, Y Wang, Y Wang, G Cui and Y Shen, Optik 127, 4136 (2016)

    Article  ADS  Google Scholar 

  16. B Zhang and F Deng, Nonlinear Dyn. 77, 1519 (2014)

    Article  Google Scholar 

  17. J Sun, S Jiang, G Cui and Y Wang, J. Comput. Nonlinear Dyn. 11, 034501 (2016)

    Article  Google Scholar 

  18. Y Yu, Chaos Solitons Fractals 33, 1197 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  19. P He, C G Jing, T Fan and C Z Chen, Complexity 19, 10 (2014)

    Article  MathSciNet  Google Scholar 

  20. M Peil, M Jacquot, Y K Chembo, L Larger and T Erneux, Phys. Rev. E 79, 026208 (2009)

    Article  ADS  Google Scholar 

  21. X Wang, H Yu, S Zhong and R P Agarwal, Appl. Math. Model. 34, 3850 (2010)

    Article  MathSciNet  Google Scholar 

  22. J Lu, D W C Ho and J Cao, Int. J. Bifurc. Chaos 18, 3101 (2008)

    Article  Google Scholar 

  23. Q Song and J Cao, Adv. Differ. Equ. 2011, 16 (2011)

    Article  Google Scholar 

  24. Y Wang and J Cao, Nonlinear Anal. Real World Appl. 14, 842 (2013)

    Article  MathSciNet  Google Scholar 

  25. J Cao, A Alofi, A Al-Mazrooei and A Elaiw, Abstr. Appl. Anal. 2013, 940573 (2013)

    Google Scholar 

  26. C F Feng, Y Zhang, J T Sun, W Qi and Y H Wang, Chaos Solitons Fractals 38, 743 (2008)

    Article  ADS  Google Scholar 

  27. J Cao, D W C Ho and Y Yang, Phys. Lett. A 373, 3128 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  28. S P Ansari and S Das, Math. Methods Appl. Sci. 38, 726 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  29. A Khan and Shikha, Pramana – J. Phys. 88, 91 (2017)

    Article  ADS  Google Scholar 

  30. V K Yadav, G Prasad, T Som and S Das, Chin. J. Phys. 55, 457 (2017)

    Article  Google Scholar 

  31. J K Hale, Theory of functional differential equations (Academic Press, New York, 1977)

    Book  MATH  Google Scholar 

  32. N N Krasovskii and J L Brenner, Stability of motion: Applications of Lyapunov’s second method to differential systems and equations with delay (Stanford University Press, California, 1963)

    Google Scholar 

  33. M M El-Dessoky, M T Yaseen and E S Aly, Appl. Math. Comput. 243, 283 (2014)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jamia Millia Islamia, Delhi, 110 025, India

    Ayub Khan

  2. Department of Mathematics, Shivaji College, Delhi, 110 027, India

    Mridula Budhraja

  3. Department of Mathematics, University of Delhi, Delhi, 110 007, India

    Aysha Ibraheem

Authors
  1. Ayub Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mridula Budhraja
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aysha Ibraheem
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aysha Ibraheem.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, A., Budhraja, M. & Ibraheem, A. Combination–combination synchronisation of time-delay chaotic systems for unknown parameters with uncertainties and external disturbances. Pramana - J Phys 91, 20 (2018). https://doi.org/10.1007/s12043-018-1588-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-018-1588-z

Keywords

PACS Nos

Search

Navigation