Skip to main content
SpringerLink
Log in

Chaotic dynamics of the fractional order nonlinear system with time delay

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

This paper presents the fractional order model of a nonlinear autonomous continuous-time difference-differential equation with only one variable. Numerical simulation results of the fractional order model demonstrate the existence of chaos when system order \(q\ge 0.2\). Values of the delay time \(\tau \) in which chaotic behavior is observed at system order \(q\) are quantitatively defined using the largest Lyapunov exponents obtained from the output time series.

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

Similar content being viewed by others

References

  1. Oldham, K., Spainer, J.: Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  3. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2000)

  4. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. CAS-I 47, 25–39 (2000)

    Article  Google Scholar 

  5. Moon, F.C.: Chaotic and Fractal Dynamics. Wiley, New York (1992)

    Book  Google Scholar 

  6. Schweppe, F.C.: Uncertain Dynamic Systems. Prentice-Hall Int. Inc, Englewood Cliffs (1977)

    Google Scholar 

  7. Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. CAS-I 42, 485–490 (1995)

    Article  Google Scholar 

  8. Arena, P., Fortuna, L., Porto, D.: Chaotic behavior in noninteger-order cellular neural networks. Phys. Rev. E 61, 776–781 (2000)

    Article  Google Scholar 

  9. Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fract. 16, 339–351 (2003)

    Article  MATH  Google Scholar 

  10. Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A 341, 55–61 (2004)

    Google Scholar 

  11. Li, C., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fract. 22, 443–450 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, C., Chen, G.: Chaos in the fractional-order Chen system and its control. Chaos Solitons Fract. 22, 549–554 (2004)

    Article  MATH  Google Scholar 

  13. Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M.: Chaos in a new system with fractional order. Chaos Solitons Fract. 31, 1203–1212 (2007)

    Google Scholar 

  14. Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Solitons Fract. 27, 985–988 (2006)

    Google Scholar 

  15. Jun-Guo, L.: Chaotic dynamics of the fractional-order Ikeda delay system and its synchronization. Chin. Phys. 15, 301–305 (2006)

    Google Scholar 

  16. Wang, D., Yu, J.: Chaos in the Fractional Order Logistic Delay System, pp. 646–651. ICCCAS, China (2008)

  17. Zhu, H., Zhou, S., Zhang, W.: Chaos and Synchronization of Time-Delayed Fractional Neuron Network System, pp. 2937–2941. ICYCS, China (2008)

  18. Çelik, V., Demir, Y.: Chaotic fractional order delayed cellular neural network. In: Baleanu, D., Guvenc, Z.B., Tenreiro Machado, J.A. (eds.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer, 313–320 (2010)

  19. Uçar, A., Bishop, S.R.: Chaotic behaviour in a nonlinear delay system. Int. J. Nonlinear Sci. Num. Sim. 2, 289–294 (2001)

    MATH  Google Scholar 

  20. Uçar, A.: A prototype model for chaos studies. Int. J. Eng. Sci. 40, 251–258 (2002)

    Article  MATH  Google Scholar 

  21. Uçar, A.: On the chaotic behavior of a prototype delayed dynamical system. Chaos Solitons Fract. 16, 187–194 (2003)

    Article  MATH  Google Scholar 

  22. Bhalekar, S.: Dynamical analysis of fractional order Uçar prototype delayed system. Signal Image Video Process. 6, 513–519 (2012)

    Article  Google Scholar 

  23. Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37, 1465–1470 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. 265, 229–248 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sun, K., Wang, X., Sprott, J.C.: Bifurcations and chaos in fractional-order simplified Lorenz system. Int. J. Bifurcation Chaos 20, 1209–1219 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlinear Dyn. 38, 117–131 (2004)

    Article  MATH  Google Scholar 

  28. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from time series. Phys. D 16, 285–317 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65, 117–134 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hilborn, R.C.: Chaos and Nonlinear Dynamics. Oxford University Press, New York (2000)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Electronics Engineering, Faculty of Engineering, Firat University, 23119 , Elazig, Turkey

    Vedat Çelik & Yakup Demir

Authors
  1. Vedat Çelik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yakup Demir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vedat Çelik.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Çelik, V., Demir, Y. Chaotic dynamics of the fractional order nonlinear system with time delay. SIViP 8, 65–70 (2014). https://doi.org/10.1007/s11760-013-0461-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-013-0461-2

Keywords

Search

Navigation