A general robust method for the synchronization of fractional-integer-order 3-D continuous-time quadratic systems

  • Hannachi Fareh Email author


In this paper, a general robust synchronization method for fractional-order and integer-order 3-D continuous-time quadratic systems is introduced. Based on the idea of the decomposition of the controller in the response system in two sub-controllers and the stability theory of the linear integer-order system, we design the effective controller to achieve synchronization between fractional-order and integer-order 3-D quadratic continuous-time systems. Finally, the fractional-order Lü system and the Rössler system of integer order are used to demonstrate the effectiveness of the presented method with numerical simulation.


Fractional-order system Integer-order system Synchronization 3-D chaotic system Chaotic attractor 

Mathematics Subject Classification

26A33 34D20 37D45 37C75 


  1. 1.
    Pecora L, Carroll T (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Grebogi C, Lai YC (1997) Controlling chaotic dynamical systems. Sys Control Lett 31:307–312MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ho MC, Hung YC (2002) Synchronization of two different systems by using generalized active control. Phys Lett A 301:424–428MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Elhadj Z, Sprott JC (2012) A universal nonlinear control law for the synchronization of arbitrary 3-D continuous-time quadratic systems. Adv Sys Sci Appl 12:347–352Google Scholar
  5. 5.
    Ahmad I, Saaban A, Ibrahin A, Shahzad M (2014) A research on the synchronization of two novel chaotic systems based on a nonlinear active control algorithm. Eng Tech Appl Sci Res 5:739–747Google Scholar
  6. 6.
    Sun J, Shen Y, Wang X et al (2014) Finitetime combination-combination synchronization of four dierent chaotic systems with unknown parameters via sliding mode control. Nonlinear Dyn 76:383–397CrossRefzbMATHGoogle Scholar
  7. 7.
    Wu X, Lu J (2003) Parameter identification and backstepping control of uncertain Lu system. Chaos Solitons Fract 18:721–729MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Adloo H, Roopaei M (2011) Review article on adaptive synchronization of chaotic systems with unknown parameters. Nonlinear Dyn 65:141–159MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zimmermann HJ (ed) (1996) Fuzzy control. In: Fuzzy set theory-and its applications. Springer, DordrechtGoogle Scholar
  10. 10.
    Wang F, Liu C (2007) Synchronization of unified chaotic system based on passive control. Phy D Nonlinear Phen 225:55–60MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mainieri R, Rehacek J (1999) Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 82:3042CrossRefGoogle Scholar
  12. 12.
    Du H, Zeng Q, Wang C (2008) Function projective synchronization of different chaotic systems with uncertain parameters. Phys Lett A 372:5402–5410CrossRefzbMATHGoogle Scholar
  13. 13.
    Petráš I (ed) (2011) Fractional-order chaotic systems. In: Fractional-order nonlinear systems. Springer, BerlinGoogle Scholar
  14. 14.
    Lu JG, Chen G (2006) A note on the fractional-order Chen system. Chaos Solitons Fract 27:685–688CrossRefzbMATHGoogle Scholar
  15. 15.
    Li C, Chen G (2004) Chaos in the fractional order Chen system and its control. Chaos Solitons Fract 22:549–554CrossRefzbMATHGoogle Scholar
  16. 16.
    Li C, Chen G (2004) Chaos and hyperchaos in the fractional-order Rössler equations. Phys A Stat Mechan Appl 341:55–61CrossRefGoogle Scholar
  17. 17.
    Coronel-Escamilla A, Gómez-Aguilar JF, López-López MG, Alvarado-Martínez VM, Guerrero-Ramírez GV (2016) Triple pendulum model involving fractional derivatives with different kernels. Chaos Solitons Fract 91:248–261MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Coronel-Escamilla A, Gómez-Aguilar JF, Torres L, Escobar-Jiménez RF, Valtierra-Rodríguez M (2017) Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order. Phys A-Stat Mechan Appl 487:1–21MathSciNetCrossRefGoogle Scholar
  19. 19.
    Owolabi KM, Atangana A (2017) Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann–Liouville sense. Chaos Solitons Fract 99:171–179MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zuñiga-Aguilar CJ, Gómez-Aguilar JF, Escobar-Jiménez RF, Romero-Ugalde HM (2018) Robust control for fractional variable-order chaotic systems with non-singular kernel. Europ Phys J Plus 133:13CrossRefGoogle Scholar
  21. 21.
    Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, AmsterdamzbMATHGoogle Scholar
  22. 22.
    Ouannas A, Abdelmalek S, Bendoukha S (2017) Coexistence of some chaos synchronization types in fractional-order differential equations. Electron J Differ Eq 2017:1804–1812MathSciNetzbMATHGoogle Scholar
  23. 23.
    Chen D, Wu C, Iu HH, Ma X (2013) Circuit simulation for synchronization of a fractional-order and integer-order chaotic system. Nonlinear Dyn 73:1671–1686MathSciNetCrossRefGoogle Scholar
  24. 24.
    Xu Y, Wang H, Li Y, Pei B (2014) Image encryption based on synchronization of fractional chaotic systems. Commun Nonlinear Sci Numer Simul 19:3735–3744MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sheu LJ (2011) A speech encryption using fractional chaotic systems. Nonlinear Dyn 65:103–108MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yin C, Dadras S, Zhong SM, Chen Y (2013) Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach. Appl Math Model 37:2469–2483MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yin C, Zhong SM, Chen WF (2012) Design of sliding mode controller for a class of fractional-order chaotic systems. Commun Nonlinear Sci Numer Simul 17:356–366MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Chen DY, Liu YX, Ma XY, Zhang RF (2012) Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn 67:893–901MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Bhalekar S, Daftardar-Gejji V (2010) Synchronization of different fractional order chaotic systems using active control. Commun Nonlinear Sci Numer Simul 15:3536–3546CrossRefzbMATHGoogle Scholar
  30. 30.
    Agrawal SK, Srivastava M, Das S (2012) Synchronization of fractional order chaotic systems using active control method. Chaos Solitons Fract 45:737–752CrossRefGoogle Scholar
  31. 31.
    Zhou P, Zhu W (2011) Function projective synchronization for fractional-order chaotic systems. Nonlinear Anal Real 12:811–816MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang X, Zhang X, Ma C (2012) Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dyn 69:511–517MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang S, Yu Y, Diao M (2010) Hybrid projective synchronization of chaotic fractional order systems with different dimensions. Phys A Stat Mechan Appl 389:4981–4988CrossRefGoogle Scholar
  34. 34.
    Chen D, Zhang R, Sprott JC, Ma X (2012) Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control. Nonlinear Dyn 70:1549–1561MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gang-Quan S, Zhi-Yong S, Yan-Bin Z (2011) A general method for synchronizing an integer-order chaotic system and a fractional-order chaotic system. Chinese Phys B 20:080505CrossRefGoogle Scholar
  36. 36.
    Yang LX, He WS, Liu XJ (2011) Synchronization between a fractional-order system and an integer order system. Comput Math App 62:4708–4716MathSciNetzbMATHGoogle Scholar
  37. 37.
    Chen D, Zhang R, Sprott JC, Chen H, Ma X (2012) Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control. Chaos 22:023130MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hahn W (1967) Stability of motion. Springer, BerlinCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Managment SciencesUniversity of TebessaTebessaAlgeria

Personalised recommendations