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2016 | OriginalPaper | Buchkapitel

Synchronization of Chaotic Dynamical Systems in Discrete-Time

verfasst von : Adel Ouannas, M. Mossa Al-sawalha

Erschienen in: Advances in Chaos Theory and Intelligent Control

Verlag: Springer International Publishing

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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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Literatur
1.
Zurück zum Zitat Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design, Studies in computational intelligence. Springer Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design, Studies in computational intelligence. Springer
2.
Zurück zum Zitat Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control, Studies in computational intelligence. Springer Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control, Studies in computational intelligence. Springer
3.
Zurück zum Zitat 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 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
4.
Zurück zum Zitat Blasius B, Stone L (2000) Chaos and phase synchronization in ecological systems. Inter Inter J Bifur Chaos 10:2361–2380 Blasius B, Stone L (2000) Chaos and phase synchronization in ecological systems. Inter Inter J Bifur Chaos 10:2361–2380
5.
Zurück zum Zitat Lakshmanan M, Murali K (1996) Chaos in nonlinear oscillators: controlling and synchronization. Singapore Lakshmanan M, Murali K (1996) Chaos in nonlinear oscillators: controlling and synchronization. Singapore
6.
Zurück zum Zitat Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations, Studies in fuzziness and soft computing. Springer Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations, Studies in fuzziness and soft computing. Springer
7.
Zurück zum Zitat Boccaletti S, Kurths J, Osipov G, Valladares D, Zhou C (2002) The synchronization of chaotic systems. Phys Rep 1–101 Boccaletti S, Kurths J, Osipov G, Valladares D, Zhou C (2002) The synchronization of chaotic systems. Phys Rep 1–101
8.
Zurück zum Zitat Pecora L, Carrol T (1990) Synchronization in chaotic systems. Phys Rev Lett 821–824 Pecora L, Carrol T (1990) Synchronization in chaotic systems. Phys Rev Lett 821–824
9.
Zurück zum Zitat Grassi G (2013) Continuous-time chaotic systems: arbitrary full-state hybrid projective synchronization via a scalar signal. Chin Phys B 22:080506–6 Grassi G (2013) Continuous-time chaotic systems: arbitrary full-state hybrid projective synchronization via a scalar signal. Chin Phys B 22:080506–6
10.
Zurück zum Zitat Hui LX, Yu SM (2011) Adaptive full state hybrid function projective lag synchronization in chaotic continuous-time system. Adv Mater Res 383–390 Hui LX, Yu SM (2011) Adaptive full state hybrid function projective lag synchronization in chaotic continuous-time system. Adv Mater Res 383–390
11.
Zurück zum Zitat Junguo L, Yugeng X (2003) Linear generalized synchronization of continuous-time chaotic systems. Chaos Solitons Fract 825–831 Junguo L, Yugeng X (2003) Linear generalized synchronization of continuous-time chaotic systems. Chaos Solitons Fract 825–831
12.
Zurück zum Zitat 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 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
13.
Zurück zum Zitat Ouannas A (2014) Chaos synchronization approach based on new criterion of stability. Nonlinear Dyn Syst Theo 395–401 Ouannas A (2014) Chaos synchronization approach based on new criterion of stability. Nonlinear Dyn Syst Theo 395–401
14.
Zurück zum Zitat 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 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
15.
Zurück zum Zitat 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 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
16.
Zurück zum Zitat 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 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
17.
Zurück zum Zitat 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 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
18.
Zurück zum Zitat 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 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
19.
Zurück zum Zitat 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 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
20.
Zurück zum Zitat 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 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
21.
Zurück zum Zitat Ju H (2007) A New approach to synchronization of discrete-time chaotic systems. J Phys Soc Jpn 093,002–093,013 Ju H (2007) A New approach to synchronization of discrete-time chaotic systems. J Phys Soc Jpn 093,002–093,013
22.
Zurück zum Zitat Strogatz S (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, Studies In Nonlinearity Strogatz S (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, Studies In Nonlinearity
23.
Zurück zum Zitat Matsumoto T, Chua L, Kobayashi K (1986) Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circuits Syst 11:1143–1147 Matsumoto T, Chua L, Kobayashi K (1986) Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circuits Syst 11:1143–1147
24.
Zurück zum Zitat Stoop R, Peinke J, Röhricht B, übener RH (1989) A p-Ge semiconductor experiment showing chaos and hyperchaos. Phys D 35:4352–4425 Stoop R, Peinke J, Röhricht B, übener RH (1989) A p-Ge semiconductor experiment showing chaos and hyperchaos. Phys D 35:4352–4425
25.
Zurück zum Zitat Eduardo L, Ruiz-Herrera A (2012) Chaos in discrete structured population models. SIAM J Appl Dynam Syst 11:1200–1214 Eduardo L, Ruiz-Herrera A (2012) Chaos in discrete structured population models. SIAM J Appl Dynam Syst 11:1200–1214
26.
Zurück zum Zitat Zhang W (2006) Discrete dynamical systems, bifurcations, and chaos in economics. Elsevier, Boston Zhang W (2006) Discrete dynamical systems, bifurcations, and chaos in economics. Elsevier, Boston
27.
Zurück zum Zitat Hénon M (1976) A two dimentionnal mapping with a strange attractor. Comm Math Phys 50:69–76 Hénon M (1976) A two dimentionnal mapping with a strange attractor. Comm Math Phys 50:69–76
28.
Zurück zum Zitat Lozi R (1978) Un Attracteur etrange du Type Attracteur de Henon. J Phys 9:9–10 Lozi R (1978) Un Attracteur etrange du Type Attracteur de Henon. J Phys 9:9–10
29.
Zurück zum Zitat Itoh M, Yang T, Chua L (2001) Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int J Bifurcat Chaos 551–560 Itoh M, Yang T, Chua L (2001) Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int J Bifurcat Chaos 551–560
30.
Zurück zum Zitat Zeraoulia E, Sprott JC (2009) The discrete hyperchaotic double scroll. Int J Bifurcat Chaos 19:1023–1027 Zeraoulia E, Sprott JC (2009) The discrete hyperchaotic double scroll. Int J Bifurcat Chaos 19:1023–1027
31.
Zurück zum Zitat Stefanski K (1998) Modelling chaos and hyperchaos with 3D maps. Chaos Solitons Fractals 9:83–93 Stefanski K (1998) Modelling chaos and hyperchaos with 3D maps. Chaos Solitons Fractals 9:83–93
32.
Zurück zum Zitat Hitzl D, Zele F (1985) An exploration of the Hénon quadratic map. Phys D 305–326 Hitzl D, Zele F (1985) An exploration of the Hénon quadratic map. Phys D 305–326
33.
Zurück zum Zitat Baier G, Klein M (1990) Maximum hyperchaos in generalized Hénon maps. Phys Lett A 51:281–284 Baier G, Klein M (1990) Maximum hyperchaos in generalized Hénon maps. Phys Lett A 51:281–284
34.
Zurück zum Zitat Wang X (2003) Chaos in complex nonlinear systems publishing house of electronics industry. Beijing Wang X (2003) Chaos in complex nonlinear systems publishing house of electronics industry. Beijing
35.
Zurück zum Zitat 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 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
36.
Zurück zum Zitat Bustos A, Hernández C (2009) Synchronization of discrete-time hyperchaotic systems: an apllication in communications. Chaos Solitons Fract 41:1301–1310 Bustos A, Hernández C (2009) Synchronization of discrete-time hyperchaotic systems: an apllication in communications. Chaos Solitons Fract 41:1301–1310
37.
Zurück zum Zitat Bustos A, Hernández C, Gutiérrez CL, Castillo P (2008) Synchronization of different hyperchaotic maps for encryption. Nonlinear Dyn Sys Theo 8:221–236 Bustos A, Hernández C, Gutiérrez CL, Castillo P (2008) Synchronization of different hyperchaotic maps for encryption. Nonlinear Dyn Sys Theo 8:221–236
38.
Zurück zum Zitat FilaliR., BenrejebM., BorneP. (2014) On observer-based secure communication design using discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simulat 13:1424–1432 FilaliR., BenrejebM., BorneP. (2014) On observer-based secure communication design using discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simulat 13:1424–1432
39.
Zurück zum Zitat González I, Hernandez C (2013) Double hyperchaotic encryption for security in biometric systems. Nonlinear Dyn Sys Theo 13:55–68 González I, Hernandez C (2013) Double hyperchaotic encryption for security in biometric systems. Nonlinear Dyn Sys Theo 13:55–68
40.
Zurück zum Zitat 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 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
41.
Zurück zum Zitat Liu W, Wang Z, Zhang W (2012) Controlled synchronization of discrete-time chaotic systems under communication constraints. Nonlinear Dyn 69:223–230 Liu W, Wang Z, Zhang W (2012) Controlled synchronization of discrete-time chaotic systems under communication constraints. Nonlinear Dyn 69:223–230
42.
Zurück zum Zitat Lu J, Xi Y (2005) Chaos communication based on synchronization of discrete-time chaotic systems. Chin Phys 14:274–278 Lu J, Xi Y (2005) Chaos communication based on synchronization of discrete-time chaotic systems. Chin Phys 14:274–278
43.
Zurück zum Zitat Solak E (2005) Cryptanalysis of observer based discrete-time chaotic encryption schemes. Inter J Bifurcat Chaos 15:653–658 Solak E (2005) Cryptanalysis of observer based discrete-time chaotic encryption schemes. Inter J Bifurcat Chaos 15:653–658
44.
Zurück zum Zitat 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 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
45.
Zurück zum Zitat 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 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
46.
Zurück zum Zitat 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 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
47.
Zurück zum Zitat Li Y, Chen Y, Li B (2009) Adaptive Function Projective Synchronization of Discrete-Time Chaotic Systems. Chin Phys Lett 26:040, 504–4 Li Y, Chen Y, Li B (2009) Adaptive Function Projective Synchronization of Discrete-Time Chaotic Systems. Chin Phys Lett 26:040, 504–4
48.
Zurück zum Zitat Chai Y, Lü L, Zhao H (2010) Lag Synchronization between discrete chaotic systems with diverse structure. Math Mech-Engl 31:733–738 Chai Y, Lü L, Zhao H (2010) Lag Synchronization between discrete chaotic systems with diverse structure. Math Mech-Engl 31:733–738
49.
Zurück zum Zitat Jianfeng L (2008) Generalized (complete, lag, anticipated) synchronization of discrete-time chaotic systems. Commun Nonlinear Sci Numer Simulat 13:1851–1859 Jianfeng L (2008) Generalized (complete, lag, anticipated) synchronization of discrete-time chaotic systems. Commun Nonlinear Sci Numer Simulat 13:1851–1859
50.
Zurück zum Zitat 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 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
51.
Zurück zum Zitat 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 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
52.
Zurück zum Zitat 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 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
53.
Zurück zum Zitat Ouannas A (2014) On full-state hybrid projective synchronization of general discrete chaotic systems. J Nonlinear Dyn 17:1–6 Ouannas A (2014) On full-state hybrid projective synchronization of general discrete chaotic systems. J Nonlinear Dyn 17:1–6
54.
Zurück zum Zitat An H, Chen Y (2009) The function cascade synchronization scheme for discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simulat 14:1494–1501 An H, Chen Y (2009) The function cascade synchronization scheme for discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simulat 14:1494–1501
55.
Zurück zum Zitat Grassi G (2012) Generalized synchronization between different chaotic maps via dead-beat control. Chin Phys B 21:050505–6 Grassi G (2012) Generalized synchronization between different chaotic maps via dead-beat control. Chin Phys B 21:050505–6
56.
Zurück zum Zitat Ma Z, Liu Z, Zhang G (2007) Generalized synchronization of discrete systems. Appl Math Mech 28:609–614 Ma Z, Liu Z, Zhang G (2007) Generalized synchronization of discrete systems. Appl Math Mech 28:609–614
57.
Zurück zum Zitat 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 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
58.
Zurück zum Zitat Chai Y, Chen L, Wu R (2012) Inverse projective synchronization between two different hyperchaotic systems with fractional order. J Appl Math 1–18 Chai Y, Chen L, Wu R (2012) Inverse projective synchronization between two different hyperchaotic systems with fractional order. J Appl Math 1–18
59.
Zurück zum Zitat Ouannas A, Mahmoud E (2014) Inverse matrix projective synchro-nization for discrete chaotic systems with different dimensions. Intell Electron Syst 3:188–192 Ouannas A, Mahmoud E (2014) Inverse matrix projective synchro-nization for discrete chaotic systems with different dimensions. Intell Electron Syst 3:188–192
60.
Zurück zum Zitat 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 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
61.
Zurück zum Zitat Ouannas A, Odibat Z (2015) Generalized synchronization of different dimensional chaotic dynamical systems in discrete time. Nonlinear Dyn 81:765–771 Ouannas A, Odibat Z (2015) Generalized synchronization of different dimensional chaotic dynamical systems in discrete time. Nonlinear Dyn 81:765–771
Metadaten
Titel
Synchronization of Chaotic Dynamical Systems in Discrete-Time
verfasst von
Adel Ouannas
M. Mossa Al-sawalha
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-30340-6_5