Generalized Multi-synchronization of Fractional Order Liouvillian Chaotic Systems Using Fractional Dynamical Controller

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


The attempt to understand the synchronize of a pair of systems have been extended to the study of a more complex problem involving multiple systems, of course, motivated by problems in the integer order case where synchronization is observed such as rendezvous, formation control, flocking and schooling, attitude alignment, sensor networks, distributed computing, consensus, and complex networks in general Florian Dörfler and Francesco Bullo, Automatica, 50(6):1539–1564, 2014, [1], Strogatz, Nature, 410(6825):268–276, 2001, [6], Martínez-Guerra et al., Applied Mathematics and Computation, 282:226–236, 2016, [4], Hale, J. Dyn. Diff. Eqns. 9(1):1–52, 1997, [2], Lin and Wang, Fuzzy Sets and Systems, 161(15):2066–2080, 2010, [3], Wang et al., IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40(6):1468–1479, 2010, [7], Reza Olfati Saber et al., Proceedings of the IEEE, 95(1):215–233, 2007, [5], Wei Ren et al., IEEE Control Systems Magazine, 72–81, 2007, [8].


  1. 1.
    Florian Dörfler; Francesco Bullo. Synchronization in complex networks of phase oscillators: A survey. Automatica, 2014; 50(6): 1539–1564.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. K. Hale, Diffusive coupling, dissipation, and synchronization, J. Dyn. Diff. Eqns., 1997; 9(1): 1–52.ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Lin, D., & Wang, X., Observer-based decentralized fuzzy neural sliding mode control for interconnected unknown chaotic systems via network structure adaptation. Fuzzy Sets and Systems, 2010; 161(15), 2066–2080.ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Martínez-Guerra, C. D. Cruz-Ancona, C. D., & C. A. Pérez-Pinacho, Generalized multi-synchronization viewed as a multi-agent leader-following consensus problem, Applied Mathematics and Computation, 2016; 282: 226–236.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Reza Olfati Saber, Alex Fax, Richard M. Murray, Consensus and Cooperation in Networked Multi-Agent Systems. Proceedings of the IEEE, 2007; 95(1): 215–233.CrossRefGoogle Scholar
  6. 6.
    S. H. Strogatz, Exploring complex networks, Nature, 2001; 410(6825), 268–276.ADSCrossRefGoogle Scholar
  7. 7.
    Wang, Y., Zhang, H., Wang, X., & Yang, D., Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 2010; 40(6), 1468–1479.CrossRefGoogle Scholar
  8. 8.
    Wei Ren, Randal W. Beard and M. Atkins, Information Consensus in Multivehicle Cooperative Control, IEEE Control Systems Magazine, pp. 72–81 ,2007.Google Scholar
  9. 9.
    Delshad, S. S., Asheghan, M. M., & Beheshti, M. H., Synchronization of N-coupled incommensurate fractional-order chaotic systems with ring connection. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(9): 3815–3824.ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Wang, J., Ma, Q., Chen, A., & Liang, Z., Pinning synchronization of fractional-order complex networks with Lipschitz-type nonlinear dynamics. ISA transactions, 2015, 57: 111–116.CrossRefGoogle Scholar
  11. 11.
    Wu, X., Lai, D., & Lu, H., Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes, Nonlinear Dynamics, 2012, 69(1): 667–683.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Behinfaraz, R., & Badamchizadeh, M. Optimal synchronization of two different incommensurate fractional-order chaotic systems with fractional cost function. Complexity, 2016, 21(S1): 401–416.ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Martínez-Guerra, & J. L. Mata-Machuca, Fractional generalized synchronization in a class of nonlinear fractional order systems, Nonlinear Dynamics. 2014; 77(4): 1237–1244.MathSciNetCrossRefGoogle Scholar
  14. 14.
    M.S. Tavazoei, M. Haeri, Synchronization of chaotic fractional-order systems via active sliding mode controller, Physica A: Statistical Mechanics and its Applications. 2008; 387(1): 57–70.Google Scholar
  15. 15.
    Wang, X. Y., Zhang, X., & Ma, C. Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dynamics, 2012; 69(1): 511–517.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, X. Y., He, Y. J., & Wang, M. J., Chaos control of a fractional order modified coupled dynamos system. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(12): 6126–6134.MathSciNetCrossRefGoogle Scholar
  17. 17.
    I. Petrás, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, first ed., Springer, 2011.CrossRefGoogle Scholar
  18. 18.
    R. Martínez-Guerra, J.J. Montesinos García & S.M. Delfin Prieto, Secure communications via synchronization of Liouvillian chaotic systems. Journal of the Franklin Institute, 2016; 353(17): 4384–4399.MathSciNetCrossRefGoogle Scholar
  19. 19.
    C. Aguilar-Ibañez, R. Martínez-Guerra, R. Aguilar-López, J. L. Mata-Machuca, Synchronization and parameter estimations of an uncertain Rikitake system, Physics Letters A 08/2010; 374(35): 3625–3628.ADSCrossRefGoogle Scholar
  20. 20.
    R. Martínez-Guerra & C. D. Cruz-Ancona, Algorithms of Estimation for Nonlinear Systems: A Differential and Algebraic Viewpoint, Springer 2017.CrossRefGoogle Scholar

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© 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

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