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Advances in Chaos Theory and Intelligent Control pp 101–132Cite as

Synchronization of Chaotic Dynamical Systems in Discrete-Time

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 337))

Abstract

In this study, we investigate the problem of chaos synchronization in discrete-time dynamical systems with different structures and diverse types. Based on Lyapunov stability theory, stability of lineare systems and nonlinear control methods some synchronization criterions are presented in 2D, 3D and N-dimensional discrete-time chaotic systems. Numerical examples and computer simulations are used to show the effectiveness and the feasibility of the proposed synchronization schemes.

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References

  1. Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design, Studies in computational intelligence. Springer

    Google Scholar 

  2. Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control, Studies in computational intelligence. Springer

    Google Scholar 

  3. Azar AT, Vaidyanathan S (2015) Handbook of research on advanced intelligent control engineering and automation, Advances in computational intelligence and robotics (ACIR) Book Series. Springer

    Google Scholar 

  4. Blasius B, Stone L (2000) Chaos and phase synchronization in ecological systems. Inter Inter J Bifur Chaos 10:2361–2380

    Google Scholar 

  5. Lakshmanan M, Murali K (1996) Chaos in nonlinear oscillators: controlling and synchronization. Singapore

    Google Scholar 

  6. Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations, Studies in fuzziness and soft computing. Springer

    Google Scholar 

  7. Boccaletti S, Kurths J, Osipov G, Valladares D, Zhou C (2002) The synchronization of chaotic systems. Phys Rep 1–101

    Google Scholar 

  8. Pecora L, Carrol T (1990) Synchronization in chaotic systems. Phys Rev Lett 821–824

    Google Scholar 

  9. Grassi G (2013) Continuous-time chaotic systems: arbitrary full-state hybrid projective synchronization via a scalar signal. Chin Phys B 22:080506–6

    Google Scholar 

  10. Hui LX, Yu SM (2011) Adaptive full state hybrid function projective lag synchronization in chaotic continuous-time system. Adv Mater Res 383–390

    Google Scholar 

  11. Junguo L, Yugeng X (2003) Linear generalized synchronization of continuous-time chaotic systems. Chaos Solitons Fract 825–831

    Google Scholar 

  12. Manfeng H, Zhenyuan X, Zhang R (2008) Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems. Commun Nonlinear Sci Numer Simulat 456–464

    Google Scholar 

  13. Ouannas A (2014) Chaos synchronization approach based on new criterion of stability. Nonlinear Dyn Syst Theo 395–401

    Google Scholar 

  14. Vaidyanathan S, Azar AT (2015) Analysis and control of a 4-D novel hyperchaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design, Studies in computational intelligence. Springer

    Google Scholar 

  15. Vaidyanathan S, Azar AT (2015) Analysis, control and synchronization of a nine-term 3-D novel chaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design, Studies in computational intelligence. Springer

    Google Scholar 

  16. Vaidyanathan S, Azar AT (2015) Anti-synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan-madhavan chaotic systems. In: Azar AT, Zhu, Q (eds) Advances and applications in sliding mode control systems, Studies in computational intelligence book series. Springer

    Google Scholar 

  17. Vaidyanathan S, Azar AT (2015) Hybrid synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan chaotic systems. In: Azar, AT, Zhu Q (eds) Advances and applications in sliding mode control systems, Studies in computational intelligence book series. Springer

    Google Scholar 

  18. Vaidyanathan S, Azar AT, Rajagopal K, Alexander P (2015) Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronization via active control. Int J Model Identication Control, 267–277

    Google Scholar 

  19. Vaidyanathan S, Idowu B, Azar AT (2015) Backstepping controller design for the global chaos synchronization of sprott’s jerk systems. In: Azar, AT, Vaidyanathan S (eds) Chaos modeling and control systems design, Studies in computational intelligence. Springer

    Google Scholar 

  20. Vaidyanathan S, Sampath S, Azar AT (2015) Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system. Springer

    Google Scholar 

  21. Ju H (2007) A New approach to synchronization of discrete-time chaotic systems. J Phys Soc Jpn 093,002–093,013

    Google Scholar 

  22. Strogatz S (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, Studies In Nonlinearity

    Google Scholar 

  23. Matsumoto T, Chua L, Kobayashi K (1986) Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circuits Syst 11:1143–1147

    Google Scholar 

  24. Stoop R, Peinke J, Röhricht B, übener RH (1989) A p-Ge semiconductor experiment showing chaos and hyperchaos. Phys D 35:4352–4425

    Google Scholar 

  25. Eduardo L, Ruiz-Herrera A (2012) Chaos in discrete structured population models. SIAM J Appl Dynam Syst 11:1200–1214

    Google Scholar 

  26. Zhang W (2006) Discrete dynamical systems, bifurcations, and chaos in economics. Elsevier, Boston

    Google Scholar 

  27. Hénon M (1976) A two dimentionnal mapping with a strange attractor. Comm Math Phys 50:69–76

    Google Scholar 

  28. Lozi R (1978) Un Attracteur etrange du Type Attracteur de Henon. J Phys 9:9–10

    Google Scholar 

  29. Itoh M, Yang T, Chua L (2001) Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int J Bifurcat Chaos 551–560

    Google Scholar 

  30. Zeraoulia E, Sprott JC (2009) The discrete hyperchaotic double scroll. Int J Bifurcat Chaos 19:1023–1027

    Google Scholar 

  31. Stefanski K (1998) Modelling chaos and hyperchaos with 3D maps. Chaos Solitons Fractals 9:83–93

    Google Scholar 

  32. Hitzl D, Zele F (1985) An exploration of the Hénon quadratic map. Phys D 305–326

    Google Scholar 

  33. Baier G, Klein M (1990) Maximum hyperchaos in generalized Hénon maps. Phys Lett A 51:281–284

    Google Scholar 

  34. Wang X (2003) Chaos in complex nonlinear systems publishing house of electronics industry. Beijing

    Google Scholar 

  35. Yan Z (2006) Q-S (Complete or Anticipated) Synchronization backstepping scheme in a class of discrete-time chaotic (Hyperchaotic) systems: a symbolic-numeric computation approach. Chaos 16:013,119–11

    Google Scholar 

  36. Bustos A, Hernández C (2009) Synchronization of discrete-time hyperchaotic systems: an apllication in communications. Chaos Solitons Fract 41:1301–1310

    Google Scholar 

  37. Bustos A, Hernández C, Gutiérrez CL, Castillo P (2008) Synchronization of different hyperchaotic maps for encryption. Nonlinear Dyn Sys Theo 8:221–236

    Google Scholar 

  38. FilaliR., BenrejebM., BorneP. (2014) On observer-based secure communication design using discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simulat 13:1424–1432

    Google Scholar 

  39. González I, Hernandez C (2013) Double hyperchaotic encryption for security in biometric systems. Nonlinear Dyn Sys Theo 13:55–68

    Google Scholar 

  40. Hernández C, Gutierrez L, Bustos A, Castillo P (2010) Communicating encrypted information based on synchronized hyperchaotic maps. Commun Nonlinear Sci Numer Simulat 11:337–349

    Google Scholar 

  41. Liu W, Wang Z, Zhang W (2012) Controlled synchronization of discrete-time chaotic systems under communication constraints. Nonlinear Dyn 69:223–230

    Google Scholar 

  42. Lu J, Xi Y (2005) Chaos communication based on synchronization of discrete-time chaotic systems. Chin Phys 14:274–278

    Google Scholar 

  43. Solak E (2005) Cryptanalysis of observer based discrete-time chaotic encryption schemes. Inter J Bifurcat Chaos 15:653–658

    Google Scholar 

  44. Filali R, Hammami S, Benrejeb M, Borne P (2012) On synchronization, anti-synchronization and hybrid synchronization of 3D discrete generalized Hénon map. Nonlinear Dyn Sys Theo 12:81–95

    Google Scholar 

  45. Jin Y, Xin L, Chen Y (2008) Function projective synchronization of discrete-time chaotic and hyperchaotic systems using backstepping method. Commun Theor Phys 50:111–116

    Google Scholar 

  46. Li Y, Chen Y, Li B (2009) Adaptive control and function projective synchronization in 2D discrete-time chaotic systems. Commun Theor Phys 51:270–278

    Google Scholar 

  47. Li Y, Chen Y, Li B (2009) Adaptive Function Projective Synchronization of Discrete-Time Chaotic Systems. Chin Phys Lett 26:040, 504–4

    Google Scholar 

  48. Chai Y, Lü L, Zhao H (2010) Lag Synchronization between discrete chaotic systems with diverse structure. Math Mech-Engl 31:733–738

    Google Scholar 

  49. Jianfeng L (2008) Generalized (complete, lag, anticipated) synchronization of discrete-time chaotic systems. Commun Nonlinear Sci Numer Simulat 13:1851–1859

    Google Scholar 

  50. Yin L, Tianyan D (2010) Adaptive control for anticipated function projective synchronization of 2d discrete-time chaotic systems with uncertain parameters. J Uncertain Syst 13:195–205

    Google Scholar 

  51. Yanbo G, Xiaomei Z, Guoping L, Yufan Z (2011) Impulsive synchronization of discrete-time chaotic systems under communication constraints. Commun Nonlinear Sci Numer Simulat 16:1580–1588

    Google Scholar 

  52. Grassi G, Miller DA (2012) Dead-beat full state hybrid projective synchronization for chaotic maps using a scalar synchronizing signal. Chin Phys B 17:1824–1830

    Google Scholar 

  53. Ouannas A (2014) On full-state hybrid projective synchronization of general discrete chaotic systems. J Nonlinear Dyn 17:1–6

    Google Scholar 

  54. An H, Chen Y (2009) The function cascade synchronization scheme for discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simulat 14:1494–1501

    Google Scholar 

  55. Grassi G (2012) Generalized synchronization between different chaotic maps via dead-beat control. Chin Phys B 21:050505–6

    Google Scholar 

  56. Ma Z, Liu Z, Zhang G (2007) Generalized synchronization of discrete systems. Appl Math Mech 28:609–614

    Google Scholar 

  57. Yan Z (2005) Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller. Phys Lett A 342:309–317

    Google Scholar 

  58. Chai Y, Chen L, Wu R (2012) Inverse projective synchronization between two different hyperchaotic systems with fractional order. J Appl Math 1–18

    Google Scholar 

  59. Ouannas A, Mahmoud E (2014) Inverse matrix projective synchro-nization for discrete chaotic systems with different dimensions. Intell Electron Syst 3:188–192

    Google Scholar 

  60. Ouannas A, Al-sawalha M (2015) On inverse full state hybrid pro- jective synchronization of chaotic dynamical systems in discrete-Time. Inter J Dyn Control 1–7

    Google Scholar 

  61. Ouannas A, Odibat Z (2015) Generalized synchronization of different dimensional chaotic dynamical systems in discrete time. Nonlinear Dyn 81:765–771

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, Tebessa, Algeria

    Adel Ouannas

  2. Mathematics Department, Faculty of Science, University of Hail, Hail, Kingdom of Saudi Arabia

    M. Mossa Al-sawalha

Authors
  1. Adel Ouannas
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  2. M. Mossa Al-sawalha
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Correspondence to Adel Ouannas .

Editor information

Editors and Affiliations

  1. Benha University, Benha, Egypt

    Ahmad Taher Azar

  2. Chennai, India

    Sundarapandian Vaidyanathan

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Ouannas, A., Al-sawalha, M.M. (2016). Synchronization of Chaotic Dynamical Systems in Discrete-Time. In: Azar, A., Vaidyanathan, S. (eds) Advances in Chaos Theory and Intelligent Control. Studies in Fuzziness and Soft Computing, vol 337. Springer, Cham. https://doi.org/10.1007/978-3-319-30340-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-30340-6_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30338-3

  • Online ISBN: 978-3-319-30340-6

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