In this paper, a new type of chaos synchronization in discrete-time is proposed by combining matrix projective synchronization (MPS) and generalized synchronization (GS). This new chaos synchronization type allows us to study synchronization between different dimensional discrete-time chaotic systems in different dimensions. Based on nonlinear controllers and Lyapunov stability theory, effective control schemes are introduced and new synchronization criterions are derived. Numerical simulations are used to validate the theoretical results and to verify the effectiveness of the proposed schemes.
Issue Section:
Research Papers
References
1.
Strogatz
, S. H.
, 2001
, “Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
,” Studies in Nonlinearity
, Westview Press
, Boulder.2.
Chen
, G. R.
, and Yu
, X. H.
, 2003
, Chaos Control: Theory and Applications
, Springer
, Berlin
.3.
Blasius
, B.
, and Stone
, L.
, 2000
, “Chaos and Phase Synchronization in Ecological Systems
,” Int. J. Bifur. Chaos.
, 10
(10
), pp. 2361
–2380
.10.1142/S02181274000015114.
Lakshmanan
, M.
, and Murali
, K.
, 1996
, Chaos in Nonlinear Oscillators: Controlling and Synchronization
, World Scientific
, Singapore
.5.
Han
, S. K.
, Kerrer
, C.
, and Kuramoto
, Y.
, 1995
, “Dephasing and Bursting in Coupled Neural Oscillators
,” Phys. Rev. Lett.
, 75
(17
), pp. 3190
–3193
.10.1103/PhysRevLett.75.31906.
Mengue
, A.
, and Essimbi
, B.
, 2012
, “Secure Communication Using Chaotic Synchronization in Mutually Coupled Semiconductor Lasers
,” Nonlinear Dyn.
, 70
(2
), pp. 1241
–1253
.10.1007/s11071-012-0528-67.
Pourmahmood Aghababa
, M.
, and Pourmahmood Aghababa
, H.
, 2012
, “Adaptive Finite-Time Synchronization of Non-Autonomous Chaotic Systems With Uncertainty
,” ASME J. Comput. Nonlinear Dyn.
, 8
(3
), p. 031006
.10.1115/1.40230078.
Ouannas
, A.
, 2014
, “Chaos Synchronization Approach Based on New Criterion of Stability
,” Nonlinear Dyn. Syst. Theory
, 14
(4
), pp. 395
–401
.9.
Zhang
, F.
, and Liu
, S.
, 2013
, “Full State Hybrid Projective Synchronization and Parameters Identification for Uncertain Chaotic (Hyperchaotic) Complex Systems
,” ASME J. Comput. Nonlinear Dyn.
, 9
(2
), p. 021009
.10.1115/1.402547510.
Diyi
, C.
, Weili
, Z.
, Xinzhi
, L.
, and Xiaoyi
, Ma.
, 2014
, “Synchronization and Antisynchronization of a Class of Chaotic Systems With Nonidentical Orders and Uncertain Parameters
,” ASME J. Comput. Nonlinear Dyn.
, 10
(1
), p. 011003
.10.1115/1.402771511.
Ju
, H.P.
, 2007
, “A New Approach to Synchronization of Discrete-Time Chaotic Systems
,” J. Phys. Soc. Jpn.
, 76
(9
), p. 093002
.10.1143/JPSJ.76.09300212.
Filali
, R. L.
, Hammami
, S.
, Benrejeb
, M.
, and Borne
, P.
, 2012
, “On Synchronization, Anti-Synchronization and Hybrid Synchronization of 3D Discrete Generalized Hénon Map
,” Nonlinear Dyn. Syst. Theory
, 12
(1
), pp. 81
–95
.13.
Ouannas
, A.
, 2014
, “Nonlinear Control Method of Chaos Synchronization for Arbitrary 2D Quadratic Dynamical Systems in Discrete-Time
,” Int. J. Math. Anal.
, 8
(53
), pp. 2611
–2617
.10.12988/ijma.2014.4928014.
Ouannas
, A.
, 2014
, “A New Chaos Synchronization Criterion for Discrete Dynamical Systems
,” Appl. Math. Sci.
, 8
(41
), pp. 2025
–2034
.10.12988/ams.2014.413215.
Aguilar-Bustos
, A. Y.
, and Cruz Hernandez
, Y. C.
, 2009
, “Synchronization of Discrete-Time Hyperchaotic Systems: An Application in Communications
,” Chaos Solitons Fractals
, 41
(3
), pp. 1301
–1310
.10.1016/j.chaos.2008.05.01216.
Liu
, W.
, Wang
, Z. M.
, and Zhang
, W. D.
, 2012
, “Controlled Synchronization of Discrete-Time Chaotic Systems Under Communication Constraints
,” Nonlinear. Dyn.
, 69
(1–2
), pp. 223
–230
.10.1007/s11071-011-0259-017.
Filali
, R. L.
, Benrejeb
, M.
, and Borne
, P.
, 2014
, “On Observer-Based Secure Communication Design Using Discrete-Time Hyperchaotic Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 19
(5
), pp. 1424
–1432
.10.1016/j.cnsns.2013.09.00518.
Yan
, Z. Y.
, 2005
, “Q-S Synchronization in 3D Hénon-Like Map and Generalized Hénon Map Via a Scalar Controller
,” Phys. Lett. A
, 342
(4
), pp. 309
–315
.10.1016/j.physleta.2005.04.04919.
Li
, Y.
, Chen
, Y.
, and Li
, B.
, 2009
, “Adaptive Control and Function Projective Synchronization in 2D Discrete-Time Chaotic Systems
,” Commun. Theor. Phys.
, 51
(2
), pp. 270
–278
.10.1088/0253-6102/51/2/1720.
Li
, Y.
, Chen
, Y.
, and Li
, B.
, 2009
, “Adaptive Function Projective Synchronization of Discrete-Time Chaotic Systems
,” Chin. Phys. Lett.
, 26
(4
), p. 040504
.10.1088/0256-307X/26/4/04050421.
Chai
, Y.
, Lü
, L.
, and Zhao
, H. Y.
, 2010
, “Lag Synchronization Between Discrete Chaotic Systems With Diverse Structure
,” Appl. Math. Mech. Engl.
, 31
(6
), pp. 733
–738
.10.1007/s10483-010-1307-722.
Yanbo
, G.
, Xiaomei
, Z.
, Guoping
, L.
, and Yufan
, Z.
, 2011
, “Impulsive Synchronization of Discrete-Time Chaotic Systems Under Communication Constraints
,” Commun. Nonlinear. Sci. Numer. Simul.
, 16
(3
), pp. 1580
–1588
.10.1016/j.cnsns.2010.07.00223.
Ouannas
, A.
, 2014
, “On Full-State Hybrid Projective Synchronization of General Discrete Chaotic Systems
,” J. Nonlinear Dyn.
, 2014
, p. 983293
.10.1155/2014/98329324.
Hao
, D.
, Xin
, J. L.
, and Bin
, Z. Y.
, 2012
, “Adaptive Generalized Matrix Projective Lag Synchronization Between Two Different Complex Networks With Non-Identical Nodes and Different Dimensions
,” Chin. Phys. B.
, 21
(12
), p. 120508
.10.1088/1674-1056/21/12/12050825.
Wu
, Z.
, Xu
, X.
, Chen
, G.
, and Fu
, X.
, 2014
, “Generalized Matrix Projective Synchronization of General Colored Networks With Different-Dimensional Node Dynamics
,” J. Franklin Inst.
, 351
(9
), pp. 4584
–4595
.10.1016/j.jfranklin.2014.07.00826.
Ma
, Z.
, Liu
, Z.
, and Zhang
, G.
, 2007
, “Generalized Synchronization of Discrete Systems
,” Appl. Math. Mech.
, 28
(5
), pp. 609
–614
.10.1007/s10483-007-0506-y27.
Jianfeng
, L.
, 2008
, “Generalized (Complete, Lag, Anticipated) Synchronization of Discrete-Time Chaotic Systems
,” Commun. Nonlinear. Sci. Numer. Simul.
, 13
(9
), pp. 1851
–1859
.10.1016/j.cnsns.2007.04.02228.
Grassi
, G.
, 2012
, “Generalized Synchronization Between Different Chaotic Maps Via Dead-Beat Control
,” Chin. Phys. B.
, 21
(5
), p. 050505
.10.1088/1674-1056/21/5/05050529.
Yin
, L.
, and Tianyan
, D.
, 2010
, “Adaptive Control for Anticipated Function Projective Synchronization of 2D Discrete-Time Chaotic Systems With Uncertain Parameters
,” J. Uncertain Syst.
, 4
(3
), pp. 195
–205
.30.
Yan
, Z. Y.
, 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
(1
), p. 013119
.10.1063/1.1930727Copyright © 2015 by ASME
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