Skip to main content
SpringerLink
Log in

Adaptive control and synchronization of a fractional-order chaotic system

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

In this paper, the chaotic dynamics of a three-dimensional fractional-order chaotic system is investigated. The lowest order for exhibiting chaos in the fractional-order system is obtained. Adaptive schemes are proposed for control and synchronization of the fractional-order chaotic system based on the stability theory of fractional-order dynamic systems. The presented schemes, which contain only a single-state variable, are simple and flexible. Numerical simulations are used to demonstrate the feasibility of the presented methods.

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

Figure 1
Figure 2
Figure 3
Figure 4

Similar content being viewed by others

References

  1. P L Butzer and U Westphal, An introduction to fractional calculus (World Scientific, Singapore, 2000)

    Google Scholar 

  2. P Arena, R Caponetto, L Fortuna and D Porto, Nonlinear noninteger order circuits and systems (World Scientific, Singapore, 2000)

    MATH  Google Scholar 

  3. R Hifer, Applications of fractional calculus in physics (World Scientific, New Jersey, 2001)

    Google Scholar 

  4. E Ahmed and A S Elgazzar, Physica A379, 607 (2007)

    MathSciNet  ADS  Google Scholar 

  5. A M A El-Sayed, A E M El-Mesiry and H A A El-Saka, Appl. Math. Lett. 20, 817 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. R L Bagley and R A Calico, J. Guid. Control Dyn. 14, 304 (1991)

    Article  Google Scholar 

  7. R C Koeller, Acta Mech. 58, 251 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. H H Sun, A A Abdelwahad and B Onaval, IEEE Trans. Autom. Control 29, 441 (1984)

    Article  MATH  Google Scholar 

  9. O Heaviside, Electromagnetic theory (Chelsea, New York, 1971)

    Google Scholar 

  10. B B Mandelbort, The fractal geometry of nature (Freeman, New York, 1983)

    Google Scholar 

  11. T T Hartley, C F Lorenzo and H K Qammer, IEEE Trans. Circuits Syst. I 43, 485 (1995)

    Article  Google Scholar 

  12. X Y Wang and J M Song, Commun. Nonlinear Sci. Numer. Simul. 14, 3351 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. J G Lü and G Chen, Chaos, Solitons and Fractals 27, 685 (2006)

    Article  ADS  Google Scholar 

  14. J G Lü, Phys. Lett. A354, 305 (2006)

    ADS  Google Scholar 

  15. C Li and G Chen, Phys. A: Stat. Mech. Appl. 341, 55 (2004)

    Article  Google Scholar 

  16. A S Hegazi and A E Matouk, Appl. Math. Lett. 24, 1938 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. C G Li and G R Chen, Chaos, Solitons and Fractals 22, 549 (2004)

    Article  ADS  MATH  Google Scholar 

  18. T Zhou and C P Li, Physica D212, 111 (2005)

    ADS  Google Scholar 

  19. L P Chen, Y Chai and R C Wu, Phys. Lett. A375, 2099 (2011)

    ADS  Google Scholar 

  20. L Song, J Y Yang and S Y Xu, Nonlinear Anal. 72, 2326 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. S Kuntanapreeda, Comp. Math. Appl. 63, 183 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. C Yin, S M Zhong and W F Chen, Commun. Nonlinear Sci. Numer. Simulat. 17, 356 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. M Caputo, Geophys. J. R. Astron. Soc. 13, 529 (1967)

    Article  Google Scholar 

  24. K Diethelm and N J Ford, J. Math. Anal. Appl. 265, 229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. E N Lorenz, J. Atm. Sci. 20, 130 (1963)

    Article  ADS  Google Scholar 

  26. G R Chen and T Ueta, Int. J. Bifurcat. Chaos 9, 1465 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. C X Liu, T Liu, L Liu and K Liu, Chaos, Solitons and Fractals 22, 1031 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. J H Lü and G R Chen, Int. J. Bifurcat. Chaos 12, 659 (2002)

    Article  MATH  Google Scholar 

  29. G Y Qi, G R Chen and M Wyk, Chaos, Solitons and Fractals 38, 705 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. C L Li, S M Yu and X S Luo, Acta Phys. Sin. 61, 110502 (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang, 414006, China

    CHUNLAI LI

  2. School of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang, 414006, China

    YAONAN TONG

Authors
  1. CHUNLAI LI
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YAONAN TONG
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to CHUNLAI LI.

Rights and permissions

Reprints and permissions

About this article

Cite this article

LI, C., TONG, Y. Adaptive control and synchronization of a fractional-order chaotic system. Pramana - J Phys 80, 583–592 (2013). https://doi.org/10.1007/s12043-012-0500-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12043-012-0500-5

Keywords

PACS Nos

Search

Navigation