Advertisement

International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1419–1433 | Cite as

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

  • Ayub Khan
  • NasreenEmail author
  • Lone Seth Jahanzaib
Article
  • 75 Downloads

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.

Keywords

Fractional order complex chaotic system Hybrid projective synchronization Adaptive sliding mode control technique 

Mathematics Subject Classification

37D45 37E99 37F99 37N10 

Notes

Acknowledgements

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

References

  1. 1.
    Strogatz SH (2018) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press, Boca RatonCrossRefGoogle Scholar
  2. 2.
    Koeller R (1984) Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 51(2):299–307MathSciNetCrossRefGoogle Scholar
  3. 3.
    Heaviside O (1894) Electromagnetic theoryGoogle Scholar
  4. 4.
    Shahverdiev E, Sivaprakasam S, Shore K (2002) Lag synchronization in time-delayed systems. Phys Lett A 292(6):320–324CrossRefGoogle Scholar
  5. 5.
    Mahmoud GM, Mahmoud EE (2010) Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn 62(4):875–882MathSciNetCrossRefGoogle Scholar
  6. 6.
    Mahmoud GM, Mahmoud EE (2010) Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dyn 61(1–2):141–152MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu J, Chen S, Chen L (2005) Adaptive control for anti-synchronization of Chua’s chaotic system. Phys Lett A 339(6):455–460CrossRefGoogle Scholar
  8. 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 585–593CrossRefGoogle Scholar
  9. 9.
    Mainieri R, Rehacek J (1999) Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 82(15):3042CrossRefGoogle Scholar
  10. 10.
    Khan A et al (2017) Hybrid function projective synchronization of chaotic systems via adaptive control. Int J Dyn Control 5(4):1114–1121MathSciNetCrossRefGoogle Scholar
  11. 11.
    Yang S, Duan C (1998) Generalized synchronization in chaotic systems. Chaos Solitons Fractals 9(10):1703–1707MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khan A, Bhat MA (2017) Multi-switching combination-combination synchronization of non-identical fractional-order chaotic systems. Math Methods Appl Sci 40(15):5654–5667MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ali MK, Fang JQ (1997) Synchronization of chaos and hyperchaos using linear and non-linear feedback functions. Phys Rev E 55(5):5285CrossRefGoogle Scholar
  14. 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):905–914MathSciNetCrossRefGoogle Scholar
  15. 15.
    Vaidyanathan S, Sampath S (2012) Anti-synchronization of four-wing chaotic systems via sliding mode control. Int J Autom Comput 9(3):274–279CrossRefGoogle Scholar
  16. 16.
    Khan A et al (2017) Combination synchronization of time-delay chaotic system via robust adaptive sliding mode control. Pramana 88(6):91CrossRefGoogle Scholar
  17. 17.
    Cao J, Li H, Ho DW (2005) Synchronization criteria of Lure systems with time-delay feedback control. Chaos Solitons Fractals 23(4):1285–1298MathSciNetCrossRefGoogle Scholar
  18. 18.
    Njah A (2010) Tracking control and synchronization of the new hyperchaotic liu system via backstepping techniques. Nonlinear Dyn 61(1–2):1–9MathSciNetCrossRefGoogle Scholar
  19. 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, AmsterdamzbMATHGoogle Scholar
  20. 20.
    Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 91(3):034,101CrossRefGoogle Scholar
  21. 21.
    Li C, Chen G (2004) Chaos and hyperchaos in the fractional-order rossler equations. Phys A 341:55–61MathSciNetCrossRefGoogle Scholar
  22. 22.
    Deng W, Li C (2005) Chaos synchronization of the fractional Lu-system. Phys A 353:61–72CrossRefGoogle Scholar
  23. 23.
    Wang XY, Wang MJ (2007) Dynamic analysis of the fractional-order liu system and its synchronization. Chaos Interdiscip J Nonlinear Sci 17(3):033,106CrossRefGoogle Scholar
  24. 24.
    Zhu H, Zhou S, Zhang J (2009) Chaos and synchronization of the fractional-order Chuas system. Chaos Solitons Fractals 39(4):1595–1603CrossRefGoogle Scholar
  25. 25.
    Luo C, Wang X (2013) Chaos in the fractional-order complex Lorenz system and its synchronization. Nonlinear Dyn 71(1–2):241–257MathSciNetCrossRefGoogle Scholar
  26. 26.
    Liu X, Hong L, Yang L (2014) Fractional-order complex t system: bifurcations, chaos control, and synchronization. Nonlinear Dyn 75(3):589–602MathSciNetCrossRefGoogle Scholar
  27. 27.
    Singh AK, Yadav VK, Das S (2017) Synchronization between fractional order complex chaotic systems. Int J Dyn Control 5(3):756–770MathSciNetCrossRefGoogle Scholar
  28. 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):1613–1620MathSciNetCrossRefGoogle Scholar
  29. 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):1639–1652MathSciNetCrossRefGoogle Scholar
  30. 30.
    Yau HT (2004) Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fractals 22(2):341–347MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hajipour A, Hajipour M, Baleanu D (2018) On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system. Phys A 497:139–153MathSciNetCrossRefGoogle Scholar
  32. 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 963–968Google Scholar
  33. 33.
    Vidyasagar M (2002) Nonlinear systems analysis, vol 42. Siam, New DelhiCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Jamia Millia IslamiaNew DelhiIndia

Personalised recommendations