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Synchronization on the adaptive sliding mode controller for fractional order complex chaotic systems with uncertainty and disturbances

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Abstract

The present paper purports to examine and analyse the concept of non identical complex chaotic systems of fractional order with external bounded disturbances and uncertainties. Hybrid projective synchronization has been achieved between fractional order complex Lu-system (drive system) and complex T-system (slave system). The adaptive sliding mode control technique has been used to design control law through suitable sliding surface and estimate the uncertainties and external disturbances in order to establish the stability of controlled system by using allied theorems. Also we have compared our results with prior published literature results to determine the supremacy of considered methodology. Computer simulations outcomes have established the efficacy and adeptness of the prospective scheme.

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References

  1. Strogatz SH (2018) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press, Boca Raton

    Book  Google Scholar 

  2. Koeller R (1984) Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 51(2):299–307

    Article  MathSciNet  Google Scholar 

  3. Heaviside O (1894) Electromagnetic theory

  4. Shahverdiev E, Sivaprakasam S, Shore K (2002) Lag synchronization in time-delayed systems. Phys Lett A 292(6):320–324

    Article  Google Scholar 

  5. Mahmoud GM, Mahmoud EE (2010) Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn 62(4):875–882

    Article  MathSciNet  Google Scholar 

  6. Mahmoud GM, Mahmoud EE (2010) Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dyn 61(1–2):141–152

    Article  MathSciNet  Google Scholar 

  7. Hu J, Chen S, Chen L (2005) Adaptive control for anti-synchronization of Chua’s chaotic system. Phys Lett A 339(6):455–460

    Article  Google Scholar 

  8. Vaidyanathan S, Rasappan S (2011) Hybrid synchronization of hyperchaotic Qi and Lu systems by nonlinear control. In: International conference on computer science and information technology. Springer, Berlin, pp 585–593

    Chapter  Google Scholar 

  9. Mainieri R, Rehacek J (1999) Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 82(15):3042

    Article  Google Scholar 

  10. Khan A et al (2017) Hybrid function projective synchronization of chaotic systems via adaptive control. Int J Dyn Control 5(4):1114–1121

    Article  MathSciNet  Google Scholar 

  11. Yang S, Duan C (1998) Generalized synchronization in chaotic systems. Chaos Solitons Fractals 9(10):1703–1707

    Article  MathSciNet  Google Scholar 

  12. Khan A, Bhat MA (2017) Multi-switching combination-combination synchronization of non-identical fractional-order chaotic systems. Math Methods Appl Sci 40(15):5654–5667

    Article  MathSciNet  Google Scholar 

  13. Ali MK, Fang JQ (1997) Synchronization of chaos and hyperchaos using linear and non-linear feedback functions. Phys Rev E 55(5):5285

    Article  Google Scholar 

  14. Srivastava M, Ansari S, Agrawal S, Das S, Leung A (2014) Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method. Nonlinear Dyn 76(2):905–914

    Article  MathSciNet  Google Scholar 

  15. Vaidyanathan S, Sampath S (2012) Anti-synchronization of four-wing chaotic systems via sliding mode control. Int J Autom Comput 9(3):274–279

    Article  Google Scholar 

  16. Khan A et al (2017) Combination synchronization of time-delay chaotic system via robust adaptive sliding mode control. Pramana 88(6):91

    Article  Google Scholar 

  17. Cao J, Li H, Ho DW (2005) Synchronization criteria of Lure systems with time-delay feedback control. Chaos Solitons Fractals 23(4):1285–1298

    Article  MathSciNet  Google Scholar 

  18. Njah A (2010) Tracking control and synchronization of the new hyperchaotic liu system via backstepping techniques. Nonlinear Dyn 61(1–2):1–9

    Article  MathSciNet  Google Scholar 

  19. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Elsevier, Amsterdam

    MATH  Google Scholar 

  20. Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 91(3):034,101

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Deng W, Li C (2005) Chaos synchronization of the fractional Lu-system. Phys A 353:61–72

    Article  Google Scholar 

  23. Wang XY, Wang MJ (2007) Dynamic analysis of the fractional-order liu system and its synchronization. Chaos Interdiscip J Nonlinear Sci 17(3):033,106

    Article  Google Scholar 

  24. Zhu H, Zhou S, Zhang J (2009) Chaos and synchronization of the fractional-order Chuas system. Chaos Solitons Fractals 39(4):1595–1603

    Article  Google Scholar 

  25. Luo C, Wang X (2013) Chaos in the fractional-order complex Lorenz system and its synchronization. Nonlinear Dyn 71(1–2):241–257

    Article  MathSciNet  Google Scholar 

  26. Liu X, Hong L, Yang L (2014) Fractional-order complex t system: bifurcations, chaos control, and synchronization. Nonlinear Dyn 75(3):589–602

    Article  MathSciNet  Google Scholar 

  27. Singh AK, Yadav VK, Das S (2017) Synchronization between fractional order complex chaotic systems. Int J Dyn Control 5(3):756–770

    Article  MathSciNet  Google Scholar 

  28. Cai N, Jing Y, Zhang S (2010) Modified projective synchronization of chaotic systems with disturbances via active sliding mode control. Commun Nonlinear Sci Numer Simul 15(6):1613–1620

    Article  MathSciNet  Google Scholar 

  29. Aghababa MP, Heydari A (2012) Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input non-linearities. Appl Math Model 36(4):1639–1652

    Article  MathSciNet  Google Scholar 

  30. Yau HT (2004) Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fractals 22(2):341–347

    Article  MathSciNet  Google Scholar 

  31. Hajipour A, Hajipour M, Baleanu D (2018) On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system. Phys A 497:139–153

    Article  MathSciNet  Google Scholar 

  32. Matignon D (1996) Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems applications, vol 2. IMACS, IEEE-SMC Lille, France, pp 963–968

  33. Vidyasagar M (2002) Nonlinear systems analysis, vol 42. Siam, New Delhi

    Book  Google Scholar 

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Acknowledgements

The second author is funded by the Junior Research fellowship of University Grant Commission, India ( Ref. No.: 19/06/2016(i)EU-V ).

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Authors and Affiliations

  1. Jamia Millia Islamia, New Delhi, India

    Ayub Khan,  Nasreen & Lone Seth Jahanzaib

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  1. Ayub Khan
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  2. Nasreen
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  3. Lone Seth Jahanzaib
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Correspondence to Nasreen.

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Khan, A., Nasreen & Jahanzaib, L.S. Synchronization on the adaptive sliding mode controller for fractional order complex chaotic systems with uncertainty and disturbances. Int. J. Dynam. Control 7, 1419–1433 (2019). https://doi.org/10.1007/s40435-019-00585-y

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  • DOI: https://doi.org/10.1007/s40435-019-00585-y

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