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Chaos and Hopf bifurcation control in a fractional-order memristor-based chaotic system with time delay

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Abstract.

In this paper, a time-delayed feedback controller is proposed in order to control chaos and Hopf bifurcation in a fractional-order memristor-based chaotic system with time delay. The associated characteristic equation is established by regarding the time delay as a bifurcation parameter. A set of conditions which ensure the existence of the Hopf bifurcation are gained by analyzing the corresponding characteristic equation. Then, we discuss the influence of feedback gain on the critical value of fractional order and time delay in the controlled system. Theoretical analysis shows that the controller is effective in delaying the Hopf bifurcation critical value via decreasing the feedback gain. Finally, some numerical simulations are presented to prove the validity of our theoretical analysis and confirm that the time-delayed feedback controller is valid in controlling chaos and Hopf bifurcation in the fractional-order memristor-based system.

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References

  1. L.O. Chua, IEEE Trans. Circ. Theory 18, 507 (1971)

    Article  Google Scholar 

  2. L.O. Chua, S.M. Kang, Proc. IEEE 64, 209 (1976)

    Article  MathSciNet  Google Scholar 

  3. L.O. Chua, IEEE Trans. Circ. Syst. 27, 1014 (1980)

    Article  Google Scholar 

  4. D.B. Strukov, G.S. Snider, D.R. Stewart, R.S. Williams, Nature 453, 80 (2008)

    Article  ADS  Google Scholar 

  5. J. Zha, H. Huang, T. Huang, J. Cao, A. Alsaedi, F.E. Alsaadi, Neurocomputing 267, 134 (2017)

    Article  Google Scholar 

  6. G. Velmurugan, R. Rakkiyappan, J. Cao, Neural Netw. 73, 36 (2016)

    Article  Google Scholar 

  7. Idongesit E. Ebong, Pinaki Mazumder, Proc. IEEE 100, 2050 (2012)

    Article  Google Scholar 

  8. Zhengwen Tu, Jinde Cao, Ahmed Alsaedi, Fuad Alsaadi, Neural Netw. 88, 125 (2017)

    Article  Google Scholar 

  9. S. Shin, K. Kim, S.M. Kang, IEEE Trans. Nanotechnol. 10, 266 (2011)

    Article  ADS  Google Scholar 

  10. K.S. Miller, B. Ross, An Introduction to the Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Longhorne, 1993)

  11. K.B. Oldham, J. Spanier, The Fractional Calculus, Vol. 1047 (Academic Press, New York, 1974)

  12. B.T. Krishna, K.V.V.S. Reddy, Act. Passiv. Electron. Compon. 2008, 369421 (2008)

    Article  Google Scholar 

  13. M. Nakagawa, K. Sorimachi, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 75, 1814 (1992)

    Google Scholar 

  14. S. Michio, Y. Hirano, Y.F. Miura, K. Saito, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 82, 1627 (1999)

    Google Scholar 

  15. S. Vashisth, H. Singh, A.K. Yadav, K. Singh, Opt. Int. J. Light Electron Opt. 125, 5309 (2014)

    Article  Google Scholar 

  16. F.A. Rihan, D.H. Abdelrahman, S. Lakshmanan, Appl. Math. Comput. 232, 606 (2014)

    Article  MathSciNet  Google Scholar 

  17. W.C. Chen, Chaos, Solitons Fractals 36, 1305 (2008)

    Article  ADS  Google Scholar 

  18. R. Li, Opt. Int. J. Light Electron Opt. 127, 6695 (2016)

    Article  Google Scholar 

  19. S. He, K. Sun, X. Mei, B. Yan, S. Xu, Eur. Phys. J. Plus 132, 36 (2017)

    Article  Google Scholar 

  20. V.K. Yadav, N. Srikanth, S. Das, Opt. Int. J. Light Electron Opt. 127, 10527 (2016)

    Article  Google Scholar 

  21. C. Huang, J. Cao, Physica A 473, 262 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  22. C. Huang, J. Cao, M. Xiao, A. Alsaedi, T. Hayat, Appl. Math. Comput. 292, 210 (2017)

    MathSciNet  Google Scholar 

  23. J.Z. Zhang, Z. Jin, J.R. Yan, G.Q. Sun, Nonlinear Anal. 70, 658 (2009)

    Article  MathSciNet  Google Scholar 

  24. G.Q. Sun, A. Chakraborty, Q.X. Liu, Z. Jin, K.E. Anderson, B.L. Li, Commun. Nonlinear Sci. Numer. Simul. 19, 1507 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  25. Z. Wang, X. Huang, G. Shi, Comput. Math. Appl. 62, 1531 (2011)

    Article  MathSciNet  Google Scholar 

  26. Haibo Bao, Ju H. Park, Jinde Cao, Appl. Math. Comput. 270, 543 (2015)

    Article  MathSciNet  Google Scholar 

  27. C. Huang, Y. Meng, J. Cao, A. Alsaedi, F.E. Alsaadi, Chaos, Solitons Fractals 100, 31 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  28. L. Liu, F. Pan, D. Xue, Opt. Int. J. Light Electron. 125, 7020 (2014)

    Article  Google Scholar 

  29. K. Pyragas, Phys. Lett. A 170, 421 (1992)

    Article  ADS  Google Scholar 

  30. C. Huang, J. Cao, M. Xiao, A. Alsaedi, Fuad E. Alsaadi, Appl. Math. Comput. 293, 293 (2017)

    MathSciNet  Google Scholar 

  31. Z. Liu, K.W. Chung, Int. J. Bifurc. Chaos 15, 3895 (2005)

    Article  Google Scholar 

  32. Z.S. Cheng, Neurocomputing 73, 3139 (2010)

    Article  Google Scholar 

  33. Y. Guo, W. Jiang, B. Niu, J. Frankl. Inst. 350, 155 (2013)

    Article  Google Scholar 

  34. Jihua Yang, Liqin Zhao, Chaos, Solitons Fractals 77, 332 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  35. Chengdai Huang, Jinde Cao, Min Xiao, Chaos, Solitons Fractals 87, 19 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  36. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 (Academic Press, 1998)

  37. S. Wang, Y. Yu, G. Wen, Nonlinear Anal. Hybrid Syst. 11, 129 (2014)

    Article  MathSciNet  Google Scholar 

  38. M. Itoh, L.O. Chua, Int. J. Bifurc. Chaos 18, 3183 (2008)

    Article  Google Scholar 

  39. V.T. Pham, A. Buscarino, L. Fortuna, M. Frasca, Int. J. Bifurc. Chaos 23, 1350073 (2013)

    Article  Google Scholar 

  40. W. Hu, D. Ding, Y. Zhang, N. Wang, D. Liang, Optik 130, 189 (2017)

    Article  ADS  Google Scholar 

  41. W.H. Deng, C.P. Li, J.H. Lü, Nonlinear Dyn. 48, 409 (2007)

    Article  Google Scholar 

  42. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16, 285 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  43. S. Bhalekar, V. Daftardar-Gejji, J. Fract. Calculus Appl. 1, 1 (2011) issue No. 5

    Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University, Hefei, China

    Dawei Ding, Nian Wang & Dong Liang

  2. School of Electronics and Information Engineering, Anhui University, 230601, Hefei, China

    Dawei Ding, Xin Qian, Wei Hu, Nian Wang & Dong Liang

Authors
  1. Dawei Ding
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  2. Xin Qian
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  3. Wei Hu
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  4. Nian Wang
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  5. Dong Liang
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Corresponding author

Correspondence to Nian Wang.

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Ding, D., Qian, X., Hu, W. et al. Chaos and Hopf bifurcation control in a fractional-order memristor-based chaotic system with time delay. Eur. Phys. J. Plus 132, 447 (2017). https://doi.org/10.1140/epjp/i2017-11699-9

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