Skip to main content
SpringerLink
Log in

Discrete-time chaotic systems synchronization based on vector norms approach

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

Suitable stabilization conditions obtained for continuous chaotic systems are generalized to discrete-time chaotic systems. The proposed approach, leading to these conditions for complete synchronization is based on the use of state feedback and aggregation techniques for stability studies associated with the arrow form matrix for system description. The results are successfully applied for two identical discrete-time hyper chaotic Henon maps with different orders and also for non-identical discrete-time chaotic systems with same order namely the Lozi and the Ushio maps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Louis M P, and Carroll T L, Synchronization in chaotic systems, Physical Review Letters, 1990, 64(8): 821.

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen M Y, Zhou D H, and Shang Y, A new observer-based synchronization scheme for private communication, Chaos, Solitons & Fractals, 2005, 24(4): 1025–1030.

    Article  MATH  Google Scholar 

  3. Chen J, Lu J A, and Wu X Q, Bidirectionally coupled synchronization of the generalized Lorenz systems, Journal of Systems Science and Complexity, 2011, 24(3): 433–448.

    Article  MathSciNet  MATH  Google Scholar 

  4. Lu J H, Switching control: From simple rules to complex chaotic systems, Journal of Systems Science and Complexity, 2003, 16(3): 404–413.

    MathSciNet  MATH  Google Scholar 

  5. Sonia H, Nonlinear observer design for chaotic systems synchronization with known or unknown parameters, International Journal of Emerging Science and Engineering, 2013, 1(8): 32–38.

    Google Scholar 

  6. Sonia H, Synchronization of identical chaotic systems using new state observers strategy, International Journal of Innovative Technology and Exploring Engineering, 2013, 3(2): 144–149.

    Google Scholar 

  7. Chen S H, Yang Q, and Wang C P, Impulsive control and synchronization of unified chaotic system, Chaos, Solitons & Fractals, 2004, 20(4): 751–758.

    Article  MathSciNet  MATH  Google Scholar 

  8. Xu Y H, et al., Impulsive synchronization of Lü chaotic systems via the hybrid controller, Optik-International Journal for Light and Electron Optics, 2016, 127(5): 2575–2578.

    Article  Google Scholar 

  9. Denis V E, Dynamical adaptive synchronization, International Journal of Adaptive Control and Signal Processing, 2006, 20(5): 491–507.

    MathSciNet  Google Scholar 

  10. Pai M C, Global synchronization of uncertain chaotic systems via discrete-time sliding mode control, Applied Mathematics and Computation, 2014, 227: 663–671.

    Article  MathSciNet  MATH  Google Scholar 

  11. Her-Terng Y and Shieh C S, Chaos synchronization using fuzzy logic controller, Nonlinear Analysis: Real World Applications, 2008, 9(4): 1800–1810.

    Article  MathSciNet  MATH  Google Scholar 

  12. Tian Y P, and Yu X H, Stabilizing unstable periodic orbits of chaotic systems via an optimal principle, Journal of the Franklin Institute, 2000, 337(6): 771–779.

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo S M, et al., Effective chaotic orbit tracker: A prediction-based digital redesign approach, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2000, 47(11): 1557–1570.

    Article  MathSciNet  MATH  Google Scholar 

  14. Thomas L C and Pecora L M, Synchronizing chaotic circuits, IEEE Transactions on Circuits and Systems, 1991, 38(4): 453–456.

    Article  MATH  Google Scholar 

  15. Hammami S and Mohamed B, Coexistence of synchronization and anti-synchronization for chaotic systems via feedback control, Chaotic Systems, 2011, 203.

    Google Scholar 

  16. Damon M and Grassi G, A discrete generalized hyperchaotic Henon map circuit, Proceedings of the 44th IEEE Midwest Symposium on Circuits and Systems, MWSCAS, 2001.

    Google Scholar 

  17. Zhang L G and Liu X J, The synchronization between two discrete-time chaotic systems using active robust model predictive control, Nonlinear Dynamics, 2013, 74(4): 905–910.

    Article  MathSciNet  MATH  Google Scholar 

  18. Si G Q, et al., Projective synchronization of different fractional-order chaotic systems with nonidentical orders, Nonlinear Analysis: Real World Applications, 2012, 13(4): 1761–1771.

    Article  MathSciNet  MATH  Google Scholar 

  19. Song Z, Adaptive modified function projective synchronization of unknown chaotic systems with different order, Applied Mathematics and Computation, 2012, 218(10): 5891–5899.

    Article  MathSciNet  MATH  Google Scholar 

  20. Filali R L, Benrejeb M, et al., On observer-based secure communication design using discrete-time hyperchaotic systems, Communications in Nonlinear Science and Numerical Simulation, 2014, 19(5): 1424–1432.

    Article  MathSciNet  Google Scholar 

  21. Li Z, et al., Synchronisation of discrete-time complex networks with delayed heterogeneous impulses, IET Control Theory & Applications, 2015, 9(18): 2648–2656.

    Article  MathSciNet  Google Scholar 

  22. Liu M Q, et al., H ℞ synchronization of two different discrete-time chaotic systems via a unified model, International Journal of Control, Automation and Systems, 2015, 13(1): 212–221.

    Article  MathSciNet  Google Scholar 

  23. Rihab G, Sakly A, and M’Sahli F, Different order chaotic systems synchronisation based on vector norms approach, International Journal of Modelling, Identification and Control, 2015, 24(2): 156–166.

    Article  Google Scholar 

  24. Filali, R L, et al, On synchronization, anti-synchronization and hybrid synchronization of 3D discrete generalized Hénon map, Nonlinear Dyn. Syst. Theory, 2012, 12(1): 81–95.

    MathSciNet  MATH  Google Scholar 

  25. Hammami S, Ben Saad K, and Benrejeb M, On the synchronization of identical and non-identical 4-D chaotic systems using arrow form matrix, Chaos, Solitons & Fractals, 2009, 42(1): 101–112.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sonia H, Hybrid synchronization of discrete-time hyperchaotic systems based on aggregation techniques for image encryption, Proceedings of the 14th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), IEEE, 2013.

    Google Scholar 

  27. Rulkov N F, Sushchik M M, Tsimring L S, et al., Generalized synchronization of chaos in directionally coupled chaotic systems, Physical Review E, 1995, 51(2): 980–994.

    Article  Google Scholar 

  28. Baier G and Klein M, Maximum hyperchaos in generalized Hénon maps, Physics Letters A, 1990, 151(6–7): 281–284.

    Article  MathSciNet  Google Scholar 

  29. Chen W H, Lu X M, and Zheng W X, Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks, IEEE Transactions on Neural Networks and Learning Systems, 2015, 26(4): 734–748.

    Article  MathSciNet  Google Scholar 

  30. Shi K B, et al., On designing stochastic sampled-data controller for master-slave synchronization of chaotic Lur’e system via a novel integral inequality, Communications in Nonlinear Science and Numerical Simulation, 2016, 34: 165–184.

    Article  MathSciNet  Google Scholar 

  31. Zhang L P and Jiang H B, Impulsive generalized synchronization for a class of nonlinear discrete chaotic systems, Communications in Nonlinear Science and Numerical Simulation, 2011, 16(4): 2027–2032.

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu H L and Teo K L, Stabilizability of discrete chaotic systems via unified impulsive control, Physics Letters A, 2009, 374(2): 235–240.

    Article  MATH  Google Scholar 

  33. Toshimitsu U, Chaotic synchronization and controlling chaos based on contraction mappings, Physics Letters A, 1995, 198(1): 14–22.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Industrial Systems Study and Renewable Energy (ESIER), the National Engineering School of Monastir (ENIM), University of Monastir, av.Ibn El jazzar, Skanes, 5019, Tunisia

    Rihab Gam & Anis Sakly

Authors
  1. Rihab Gam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anis Sakly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rihab Gam.

Additional information

This paper was recommended for publication by Editor DI Zengru.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gam, R., Sakly, A. Discrete-time chaotic systems synchronization based on vector norms approach. J Syst Sci Complex 30, 1012–1026 (2017). https://doi.org/10.1007/s11424-017-5267-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-017-5267-9

Keywords

Search

Navigation