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Chaos control for multi-scroll chaotic attractors generated by introducing a bipolar sigmoid function series

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Abstract

In this paper, a bipolar sigmoid function series is proposed to generate 1-D (one dimensional) and 2-D (two dimensional) multi-scroll chaotic attractors. The nonlinear dynamical behaviors of 1-D and 2-D multi-scroll chaotic systems are analyzed, including the equilibrium points, invariance, dissipation, Lyapunov exponents, fractional dimension and Poincaré map. In addition, backstepping controllers are used to stabilize the 1-D and 2-D multi-scroll chaotic systems. The results show that the 1-D and 2-D multi-scroll chaotic attractors can be generated by introducing the bipolar sigmoid function series. The mechanism for generating multi-scroll chaotic attractors is as follows: The equilibrium points with index 2 (type two) are responsible for generating scrolls, and the equilibrium points with index 1 (type one) are responsible for connecting these scrolls. The chaotic behaviors in the 1-D and 2-D multi-scroll chaotic systems are stabilized to some equilibrium points using the backstepping controllers, without determining the desired targeting orbits beforehand.

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References

  1. P Wang, J H Lü and M J Ogorzalek Neurocomputing78(1) 155 (2012).

    Google Scholar 

  2. C Li Signal Process. 118 203 (2016).

    Google Scholar 

  3. I M Ginarsa, A Soeprijanto and M H Purnomo Int. J. Electr. Power46 79 (2013).

    Google Scholar 

  4. Q X Wang, S M Yu, C Q Li, J H Lü, X L Fang, C Guyeux and J M Bahi IEEE Trans. Circuits-I 63(3) 401 (2016).

    MathSciNet  Google Scholar 

  5. L O Chua, M Komuro and T Matsumoto IEEE Trans. Circuits33(11) 1072 (1986).

    ADS  MathSciNet  Google Scholar 

  6. W Ahmad Chaos Soliton. Fract.25(3) 727 (2005).

    ADS  Google Scholar 

  7. J H Lü, S M Yu, H Leung and G R Chen IEEE Trans. Circuits-I 53(1) 149 (2006).

    ADS  Google Scholar 

  8. S M Yu, J H Lü, K S Tang and G R Chen Chaos16(3) 033126 (2006).

    ADS  Google Scholar 

  9. X M Wu and Y G He Int. J. Bifurcat. Chaos25(3) 1550041 (2015).

    Google Scholar 

  10. M Khan, T Shah, H Mahmood and M A Gondal Nonlinear Dyn. 71(3) 489 (2013).

    Google Scholar 

  11. S K Changchien, C K Huang, H H Nien and H W Shieh Chaos Soliton. Fract.39(4) 1578 (2009).

    ADS  Google Scholar 

  12. D C Mishra, R K Sharma, S Suman and A Prasad J. Inf. Secur. Appl.37 65 (2017).

    Google Scholar 

  13. X Y Zhang, S M Yu, P Chen, J H Lü, J B He and Z S Lin IEEE Trans. Circ. Syst. Vid.25(7) 1203 (2015).

    ADS  Google Scholar 

  14. J A K Suykens and J Vandewalle IEEE Trans. Circuits-I 40(11) 861 (1993).

    Google Scholar 

  15. C W Shen, S M Yu, J H Lü and G R Chen IEEE Trans. Circuits-I 61(3) 854 (2014).

    Google Scholar 

  16. M E Yalcin, J A K Suykens and J Vandewalle Int. J. Bifurcat. Chaos12(1) 23 (2002).

    Google Scholar 

  17. J H Lü, F L Han, X H Yu and G R Chen Automatica40(10) 1677 (2004).

    MathSciNet  Google Scholar 

  18. J H Lü, G R Chen, X H Yu and H Leung IEEE Trans. Circuits-I 51(12) 2476 (2004).

    MathSciNet  Google Scholar 

  19. C X Zhang and S M Yu Phys. Lett. A 374(30) 3029 (2010).

    ADS  Google Scholar 

  20. C H Wang, X W Luo and Z Wan Optik125(22) 6716 (2014).

    ADS  Google Scholar 

  21. Q H Hong, Q G Xie and P Xiao Nonlinear Dyn.87(2) 1015 (2017).

    Google Scholar 

  22. Z Chen, G L Wen, H A Zhou and J Y Chen Optik136 27 (2017).

    ADS  Google Scholar 

  23. J Han and C Moraga Int. Workshop Artif. Neural Netw. (Malga-Torremolinos, Spain) 930 195 (1995).

    Google Scholar 

  24. F Xu and P Yu J. Math. Anal. Appl.362(1) 252 (2010).

    MathSciNet  Google Scholar 

  25. Z Chen, G L Wen, H A Zhou and J Y Chen Optik130 594 (2017).

    ADS  Google Scholar 

  26. X Wang and X Li Int. J. Mod. Phys. B 24(3) 397 (2010).

    ADS  Google Scholar 

  27. O I Tacha, Ch K Volos, I M Kyprianidis, I N Stouboulos, S Vaidyanathan and V T Pham Appl. Math. Comput.276 200 (2016).

    MathSciNet  Google Scholar 

  28. U E Kocamaz and Y Uyaroglu Inf. Technol. Control43(2) 166 (2014).

    Google Scholar 

  29. X R Chen and C X Liu Nonlinear Anal-Real.11(2) 683 (2010).

    MathSciNet  Google Scholar 

  30. G Q Min, S K Duan and L D Wang Int. J. Bifurcat. Chaos26(8) 1650129 (2016).

    Google Scholar 

  31. M T Yassen Chaos Soliton. Fract.27(2) 537 (2006).

    ADS  MathSciNet  Google Scholar 

  32. A N Njah Nonlinear Dyn.61(1–2) 1 (2010).

    MathSciNet  Google Scholar 

  33. A N Njah, K S Ojo, G A Adebayo and A O Obawole Physica C 470(13–14) 558 (2010).

    ADS  Google Scholar 

  34. A Lagrioui and H Mahmoudi J. Theor. Appl. Technol.29(1) 1 (2011).

    Google Scholar 

  35. K Kemih, M Halimi, M Ghanes, H Fanit and H Salit Eur. Phys. J. Spec. Top.223(9) 1579 (2014).

    Google Scholar 

  36. U E Kocamaz, Y Uyaroglu and S Vaidyanathan Advances and Applications in Chaotic Systems (Switzerland: Springer International Publishing) Vol 636 Ch 17, pp 409 (2016).

  37. S Vaidyanathan Arch. Control Sci.27(3) 409 (2017).

    MathSciNet  Google Scholar 

  38. S Mascolo and G Grassi Phys. Rev. E 56(5) 6166 (1997).

    ADS  Google Scholar 

  39. Y Yu and S Zhang Chaos Soliton. Fract.15(5) 897 (2003).

    ADS  Google Scholar 

  40. N Jiteurtragool, T Masayoshi and W San-Um Entropy20(2) 136 (2018).

    ADS  Google Scholar 

  41. D Cafagna and G Grassi Int. J. Bifurcat. Chaos13(10) 2889 (2003).

    Google Scholar 

  42. C P Silva IEEE Trans. Circuits-I 40(10) 675 (1993).

    Google Scholar 

  43. H K Khalil Nonlinear Systems (Third Edition) (New York: Prentice Hall) Ch 8, Sec 3, pp 323 (2002).

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Acknowledgements

This work was supported by the Inner Mongolia University of Technology Foundation (Grant No. ZD201520) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2017BS0603).

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  1. Department of Automation, School of Electric Power, Jinchuan Campus, Inner Mongolia University of Technology, Hohhot, People’s Republic of China

    H. G. Jiang & M. M. Jia

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  1. H. G. Jiang
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  2. M. M. Jia
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Correspondence to M. M. Jia.

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Jiang, H.G., Jia, M.M. Chaos control for multi-scroll chaotic attractors generated by introducing a bipolar sigmoid function series. Indian J Phys 94, 851–861 (2020). https://doi.org/10.1007/s12648-019-01512-9

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