Phase and anti-phase synchronizations of fractional order hyperchaotic systems with uncertainties and external disturbances using nonlinear active control method

  • Vijay K. Yadav
  • S. K. Agrawal
  • M. Srivastava
  • S. Das


In this paper, the phase and anti-phase synchronizations between fractional order hyperchaotic Lu and 4D integral order systems with parametric uncertainties and disturbances are studied using nonlinear active control method. A new lemma is used to design the controller. Numerical simulations are presented to demonstrate the effectiveness of the method to synchronize and anti-synchronize the fractional order hyperchaotic systems. The striking feature of the article is the comparison of time of synchronization and anti-synchronization with and without the presence of uncertainties and external disturbances through graphical presentations for different particular cases.


Fractional derivative Hyperchaotic systems Nonlinear active control method Lyapunov stability Synchronization Phase-synchronization  Antiphase-synchronization 



The authors express their heartfelt thanks to the revered reviewers for their valuable suggestions for the improvement of the article. The first author acknowledges the financial support from the UGC, New Delhi, India under the JRF scheme.


  1. 1.
    Fujisaka H, Yamada T (1983) Stability theory of synchronized motion in coupled-oscillator systems. Progr Theor Phys 69:32–47MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Pecora LM, Carroll TL (1990) Synchronization in chaotic system. Phys Rev Lett 64:821–824MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen G, Dong X (1998) From chaos to order: methodologies, perspectives, and applications. World Scientific, SingaporeGoogle Scholar
  4. 4.
    Elabbssy EM, Agiza HN, El-Dessoky MM (2006) Adaptive synchronization of a hyperchaotic system with uncertain parameter. Chaos Solitons Fract 30:1133–1142MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lu J, Chen Sh (2002) Chaotic time series analysis and its application. Wuhan University Press, ChinaGoogle Scholar
  6. 6.
    Chen G, Lu J (2003) Dynamics of the Lorenz system family: analysis, control and synchronization. Beijing Scientific Press, ChinaGoogle Scholar
  7. 7.
    Li Z, Xu D (2004) A secure communication scheme using projective chaos synchronization. Chaos Solitons Fract 22:477–481Google Scholar
  8. 8.
    Wang XY (2003) Chaos in the complex nonlinearity system. Electronics Industry Press, BeijingGoogle Scholar
  9. 9.
    Li C, Chen G (2004) Chaos and hyperchaos in the fractional-order Rossler equations. Phys A 341:55–61MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cafagna D, Grassi G (2011) Observer-based synchronization for a class of fractional chaotic systems via a scalar signal: results involving the exact solution of the error dynamics. Int J Bifurc Chaos 21:955. doi: 10.1142/S021812741102874X MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cafagna D, Grassi G (2012) Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rossler systems. Nonlinear Dyn 68:117–128. doi: 10.1007/s11071-011-0208-y CrossRefMATHGoogle Scholar
  12. 12.
    Dong EZ, Chen ZQ, Yuan ZZ (2006) Synchronization of the hyperchaotic Rossler system with uncertain parameters via nonlinear control method. Optoelectron Lett 2:0389–0391CrossRefGoogle Scholar
  13. 13.
    Bhalekar S (2014) Synchronization of non-identical fractional order hyperchaotic systems using active control. World J Modell Simul 10:60–68Google Scholar
  14. 14.
    Chen M, Wu Q, Jiang C (2012) Disturbance-observer-based robust synchronization control of uncertain chaotic systems. Nonlinear Dyn 70:2421–2432. doi: 10.1007/s11071-012-0630-9 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jawaadaa W, Noorani MSM, Al-sawalha MM (2012) Robust active sliding mode anti-synchronization of hyperchaotic systems with uncertainties and external disturbances. Nonlin Anal Real World Appl 13:2403–2413Google Scholar
  16. 16.
    Kilbas A, Srivastava HM, Trujillo J (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamGoogle Scholar
  17. 17.
    Camacho NA, Manuel A, Mermoud D, Gallegos JA (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19:2951–2957Google Scholar
  18. 18.
    Pan L, Zhou W, Zhou L, Sun K (2011) Chaos synchronization between two different fractional-order hyperchaotic systems. Commun Nonlinear Sci Numer Simul 16:2628–2640MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Deng H, Li T, Wang Q, Li H (2009) A fractional-order hyperchaotic system and its synchronization. Chaos Solitons Fract 41:962–969CrossRefMATHGoogle Scholar
  20. 20.
    Cafagna D, Grassi G (2012) On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn 70:1185–1197. doi: 10.1007/s11071-012-0522-z MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Vijay K. Yadav
    • 1
  • S. K. Agrawal
    • 2
  • M. Srivastava
    • 1
  • S. Das
    • 1
  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia
  2. 2.Department of Applied SciencesBharati Vidyapeeth’s College of EngineeringDelhiIndia

Personalised recommendations