Synchronization and Anti-synchronization of Fractional Order Chaotic Systems by Means of a Fractional Integral Observer

  • Rafael Martínez-GuerraEmail author
  • Claudia Alejandra Pérez-Pinacho
Part of the Understanding Complex Systems book series (UCS)


The problem of anti-synchronization is another phenomenon of interest that occurs in chaotic oscillators. This problem has appeared in modern repetitions of Huygens’ experiments (Bennett et al., Proc: Math Phys Eng Sci 458:563–579, 2002, [2]), lasers (Uchida et al., Phys Rev A 64:023801-1–023801-6, 2001, [2]), (Wedekind and Parlitz, Int J Bifurc Chaos 11(4):1141–1147, 2001, [3]), saltwater oscillators (Nakata et al., Phys D 115:313–320, 1998, [4]), and some biological systems where a nonchaotic signal is generated (Kim et al., Phys Lett A 320:39–46, 2003, [5]). Anti-synchronization has been treated as a direct modification of synchronization, simply with a sign change in the condition required for the error, and has been attacked with methods such as the active control (Emadzadeh and Haeri, Int J Electr Comput Energ Electron Commun Eng 1(6):898–901, 2007, [6]) (Guo-Hui, Chaos, Solitons Fractals 26:87–93, 2005, [7]) and the sliding mode control (Chen et al., Nonlinear Dyn 69:35–55, 2012, [8]). It can also be induced by noise (Kawamura, Phys D 270(1):20–29, 2014, [9]).


  1. 1.
    M. Bennett, M.F. Schatz, H. Rockwood, K. Wiesenfeld, Huygens’s clocks, Proceedings: Mathematical, Physical and Engineering Sciences, 458(2019) (2002) 563–579.Google Scholar
  2. 2.
    A. Uchida, Y. Liu, I. Fischer, P. Davis, T. Aida, Chaotic antiphase dynamics and synchronization in multimode semiconductor lasers, Physical Review A 64 (2001) 023801-1–023801-6.Google Scholar
  3. 3.
    I. Wedekind, U. Parlitz, Experimental observation of synchronization and anti-synchronization of chaotic low-frequency-fluctuations in external cavity semiconductor lasers, International Journal of Bifurcation and Chaos 11(4) (2001) 1141–1147.ADSCrossRefGoogle Scholar
  4. 4.
    S. Nakata, T. Miyata, N. Ojima, K. Yoshikawa, Self-synchronization in coupled salt-water oscillators, Physica D 115 (1998) 313–320.ADSCrossRefGoogle Scholar
  5. 5.
    C.-M. Kim, S. Rim, W.-H. Kye, J.-W. Ryu, Y.-J. Park, Anti-synchronization of chaotic oscillators, Physics Letters A 320 (2003) 39–46.ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    A. A. Emadzadeh, M. Haeri, Anti-Synchronization of two Different Chaotic Systems via Active Control, International Journal of Electrical, Computer, Energetic, Electronic and Communication Engineering 1(6) (2007) 898–901.Google Scholar
  7. 7.
    L. Guo-Hui, Synchronization and anti-synchronization of Colpitts oscillators using active control, Chaos, Solitons & Fractals 26 (2005) 87–93.ADSCrossRefGoogle Scholar
  8. 8.
    D. Chen, R. Zhang, X. Ma, S. Liu, Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme, Nonlinear Dynamics 69 (2012) 35–55.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Y. Kawamura, Collective phase dynamics of globally coupled oscillators: Noise-induced anti-phase synchronization, Physica D 270(1) (2014) 20–29.ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    J.J. de Espindola, J. Neto, E. Lopes, A generalized fractional derivative approach to viscoelastic material properties measurement. Applied Mathematics and Computation 164 (2005) 493–506.CrossRefGoogle Scholar
  11. 11.
    J.J. Rosales, J. F. Gómez, M. Guía, V.I. Tkach, Fractional Electromagnetic Waves, Proceedings of the LFNM*2011 International Conference on Laser & Fiber-Optical Networks Modeling, 4-8 September 2011, Kharkov, Ukraine, p. 1–3.Google Scholar
  12. 12.
    W. Yu, Y. Luo, Y. Pi, Fractional order modeling and control for permanent magnet synchronous motor velocity servo system, Mechatronics 23 (2013) 813–820.CrossRefGoogle Scholar
  13. 13.
    J.-D. Gabano, T. Poinot, Fractional modelling and identification of thermal systems, Signal Processing 91 (2011) 531–541.CrossRefGoogle Scholar
  14. 14.
    C. Huang, J. Cao, Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system, Physica A 473 (2017) 262275.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Razminia, D. Baleanu, Complete synchronization of commensurate fractional order chaotic systems using sliding mode control, Mechatronics 23 (2013) 873–879.CrossRefGoogle Scholar
  16. 16.
    A. Razminia, V. J. Majd, D. Baleanu, Chaotic incommensurate fractional order Rössler system: active control and synchronization, Advances in Difference Equations 2011(15) (2011).CrossRefGoogle Scholar
  17. 17.
    A. Razminia, D. F. M. Torres, Control of a novel chaotic fractional order system using a state feedback technique, Mechatronics 23 (2013) 755–763.CrossRefGoogle Scholar
  18. 18.
    T. Zhou, C. Li, Synchronization in fractional-order differential systems, Physica D 212(1-2) (2005) 111–125.ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems Applications (1996) 963–968.Google Scholar
  20. 20.
    E. N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences 20 (1963) 130–141.ADSCrossRefGoogle Scholar
  21. 21.
    I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Physical Review Letters 91(3) (2003) 034101-1–034101-4.Google Scholar
  22. 22.
    O. E. Rössler, An equation for continuous chaos, Physics Letters A 57 (1976) 397–398.ADSCrossRefGoogle Scholar
  23. 23.
    C. Li, G. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Physica A 341 (2004) 55–61.ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Rafael Martínez-Guerra
    • 1
    Email author
  • Claudia Alejandra Pérez-Pinacho
    • 1
  1. 1.Automatic ControlCINVESTAV-IPNMexico CityMexico

Personalised recommendations