To synchronize quadratic chaotic systems, a synchronization scheme based on simultaneous estimation of nonlinear dynamics (SEND) is presented in this paper. To estimate quadratic terms, a compensator including Jacobian matrices in the proposed master–slave schematic is considered. According to the proposed control law and Lyapunov theorem, the asymptotic convergence of synchronization error to zero is proved. To identify unknown parameters, an adaptive mechanism is also used. Finally, a number of numerical simulations are provided for the Lorenz system and a memristor-based chaotic system to verify the proposed method.
Issue Section:
Research Papers
References
1.
Weiss
, J. N.
, Garfinkel
, A.
, Spano
, M. L.
, and Ditto
, W. L.
, 1994
, “Chaos and Chaos Control in Biology
,” J. Clin. Invest
, 93
(4
), pp. 1355
–1360
.2.
Goedgebuer
, J. P.
, Larger
, L.
, and Porte
, H.
, 1998
, “Optical Cryptosystem Based on Synchronization of Hyperchaos Generated by a Delayed Feedback Tunable Laser Diode
,” Phys. Rev. Lett.
, 80
(10
), pp. 2249
–2252
.3.
Aihara
, K.
, Takabe
, T.
, and Toyoda
, M.
, 1990
, “Chaotic Neural Networks
,” Phys. Lett. A.
, 144
(6–7
), pp. 333
–340
.4.
Schiff
, S. J.
, Jerger
, K.
, Duong
, D. H.
, Chang
, T.
, Spano
, M. L.
, and Ditto
, W. L.
, 1994
, “Controlling Chaos in the Brain
,” Nature
, 370
(6491
), pp. 615
–620
.5.
Ruiz-Herrera
, A.
, 2013
, “Chaos in Delay Differential Equations With Applications in Population Dynamics
,” Discrete. Cont. Dyn. Syst.
, 33
(4
), pp. 1633
–1644
.6.
Alvarez
, G.
, and Li
, S.
, 2006
, “Some Basic Cryptographic Requirements for Chaos-Based Cryptosystems
,” Int. J. Bifurcation Chaos
, 16
(8
), pp. 2129
–2151
.7.
Gao
, T.
, and Chen
, Z.
, 2008
, “A New Image Encryption Algorithm Based on Hyper-Chaos
,” Phys. Lett. A.
, 372
(4
), pp. 394
–400
.8.
Wang
, X. Y.
, Yang
, L.
, Liu
, R.
, and Kadir
, A.
, 2010
, “A Chaotic Image Encryption Algorithm Based on Perceptron Model
,” Nonlinear. Dyn.
, 62
(3
), pp. 615
–621
.9.
Li
, Y.
, Wang
, C.
, and Chen
, H.
, 2017
, “A Hyper-Chaos-Based Image Encryption Algorithm Using Pixel-Level Permutation and Bit-Level Permutation
,” Opt. Laser. Eng.
, 90
, pp. 238
–246
.10.
Zarei
, A.
, and Tavakoli
, S.
, 2017
, “Design and Control of a Multi-Wing Dissipative Chaotic System
,” Int. J. Dyn. Cont.
, 6
(1), 140–153.11.
Bao
, J.
, and Yang
, Q.
, 2010
, “Complex Dynamics in the Stretch-Twist-Fold Flow
,” Nonlinear. Dyn.
, 61
(4
), pp. 773
–781
.12.
Shen
, C.
, Yu
, S.
, Lu
, J.
, and Chen
, G.
, 2014
, “A Systematic Methodology for Constructing Hyperchaotic Systems With Multiple Positive Lyapunov Exponents and Circuit Implementation
,” IEEE. Trans. Circuits. Syst. I: Regular Papers.
, 61
(3
), pp. 854
–864
.13.
Chen
, L.
, Pan
, W.
, Wang
, K.
, Wu
, R.
, Machado
, J. T.
, and Lopes
, A. M.
, 2017
, “Generation of a Family of Fractional Order Hyper-Chaotic Multi-Scroll Attractors
,” Chaos, Solitons Fractals
, 105
, pp. 244
–255
.14.
Yang
, Q.
, Osman
, W. M.
, and Chen
, C.
, 2015
, “A New 6D Hyperchaotic System With Four Positive Lyapunov Exponents Coined
,” Int. J. Bifurcation Chaos
, 25
(4
), p. 1550060
.15.
Zarei
, A.
, 2015
, “Complex Dynamics in a 5-D Hyper-Chaotic Attractor With Four-Wing, One Equilibrium and Multiple Chaotic Attractors
,” Nonlinear. Dyn.
, 81
(1–2
), pp. 585
–605
.16.
Pecora
, L. M.
, and Carroll
, T. L.
, 1990
, “Synchronization in Chaotic System
,” Phys. Rev. Lett.
, 64
(8
), pp. 821
–824
.17.
Khan
, A.
, and Tyagi
, A.
, 2017
, “Hybrid Projective Synchronization Between Two Identical New 4-D Hyper-Chaotic Systems Via Active Control Method
,” Int. J. Nonlinear. Sci.
, 23
(3
), pp. 142
–150
.18.
Tavazoei
, M. S.
, and Haeri
, M.
, 2008
, “Synchronization of Chaotic Fractional-Order Systems Via Active Sliding Mode Controller
,” Phys. A.
, 387
(1
), pp. 57
–70
.19.
Chen
, M.
, Wu
, Q.
, and Jiang
, C.
, 2012
, “Disturbance-Observer-Based Robust Synchronization Control of Uncertain Chaotic Systems
,” Nonlinear. Dyn.
, 70
(4
), pp. 2421
–2432
.20.
Mohammadpour
, S.
, and Binazadeh
, T.
, 2018
, “Robust Adaptive Synchronization of Chaotic Systems With Nonsymmetric Input Saturation Constraints
,” ASME J. Comput. Nonlinear Dyn.
, 13
(1
), p. 011005
.21.
Dadras
, S.
, and Momeni
, H. R.
, 2010
, “Adaptive Sliding Mode Control of Chaotic Dynamical Systems With Application to Synchronization
,” Math. Comput. Simulat.
, 80
(12
), pp. 2245
–2257
.22.
Vasegh
, N.
, and Majd
, V. J.
, 2006
, “Adaptive Fuzzy Synchronization of Discrete-Time Chaotic Systems
,” Chaos, Solitons Fractals
, 28
(4
), pp. 1029
–1036
.23.
Wang
, H. W.
, and Gu
, H.
, 2008
, “Chaotic Synchronization in the Presence of Disturbances Based on an Orthogonal Function Neural Network
,” Asian. J. Control.
, 10
(4
), pp. 470
–477
.24.
Boutayeb
, M.
, Darouach
, M.
, and Rafaralahy
, H.
, 2002
, “Generalized State-Space Observers for Chaotic Synchronization and Secure Communication
,” IEEE. Trans. Circuits. Syst. I: Regular Papers.
, 49
(3
), pp. 345
–349
.25.
Wang
, X. Y.
, and Wang
, M. J.
, 2009
, “Chaotic Secure Communication Scheme Based on Observer
,” Commun. Nonlinear. Sci. Num. Simul.
, 14
(4
), pp. 1502
–1508
.26.
Wang
, H.
, Zhu
, X. J.
, Gao
, S. W.
, and Chen
, Z. Y.
, 2011
, “Singular Observer Approach for Chaotic Synchronization and Private Communication
,” Commun. Nonlinear. Sci. Num. Simul.
, 16
(3
), pp. 1517
–1523
.27.
Azarang
, A.
, Miri
, M.
, Kamaei
, S.
, and Asemani
, M. H.
, 2018
, “Nonfragile Fuzzy Output Feedback Synchronization of a New Chaotic System: Design and Implementation
,” ASME J. Comput. Nonlinear Dyn.
, 13
(1
), p. 011008
.28.
Wang
, P.
, Li
, D.
, Wu
, X.
, Lü
, J.
, and Yu
, X.
, 2011
, “Ultimate Bound Estimation of a Class of High Dimensional Quadratic Autonomous Dynamical Systems
,” Int. J. Bifurcation Chaos
, 21
(9
), pp. 2679
–2694
.29.
Wang
, P.
, Zhang
, Y.
, Tan
, S.
, and Wan
, L.
, 2013
, “Explicit Ultimate Bound Sets of a New Hyperchaotic System and Its Application in Estimating the Hausdorff Dimension
,” Nonlinear. Dyn.
, 74
(1–2
), pp. 133
–142
.30.
Zarei
, A.
, and Tavakoli
, S.
, 2016
, “Hopf Bifurcation Analysis and Ultimate Bound Estimation of a New 4-D Quadratic Autonomous Hyper-Chaotic System
,” Appl. Math. Comput.
, 291
, pp. 323
–339
.31.
Pogromsky
, A. Y.
, Santoboni
, G.
, and Nijmeijer
, H.
, 2003
, “An Ultimate Bound on the Trajectories of the Lorenz System and Its Applications
,” Nonlinearity.
, 16
(5
), pp. 1597
–1605
.32.
Li
, D.
, Lu
, J. A.
, Wu
, X.
, and Chen
, G.
, 2006
, “Estimating the Ultimate Bound and Positively Invariant Set for the Lorenz System and a Unified Chaotic System
,” J. Math. Anal. Appl.
, 323
(2
), pp. 844
–853
.33.
Nik
, H. S.
, Effati
, S.
, and Saberi-Nadjafi
, J. A. F. A. R.
, 2015
, “Ultimate Bound Sets of a Hyperchaotic System and Its Application in Chaos Synchronization
,” Complexity
, 20
(4
), pp. 30
–44
.34.
Li
, Q.
, Zeng
, H.
, and Li
, J.
, 2015
, “Hyperchaos in a 4D Memristive Circuit With Infinitely Many Stable Equilibria
,” Nonlinear. Dyn.
, 79
(4
), pp. 2295
–2308
.35.
Chen
, M.
, Li
, M.
, Yu
, Q.
, Bao
, B.
, Xu
, Q.
, and Wang
, J.
, 2015
, “Dynamics of Self-Excited Attractors and Hidden Attractors in Generalized Memristor-Based Chuas Circuit
,” Nonlinear. Dyn.
, 81
(1–2
), pp. 215
–226
.36.
Ma
, J.
, Chen
, Z.
, Wang
, Z.
, and Zhang
, Q.
, 2015
, “A Four-Wing Hyper-Chaotic Attractor Generated From a 4-D Memristive System With a Line Equilibrium
,” Nonlinear Dyn.
, 81
(3
), pp. 1275
–1288
.37.
Njitacke
, Z. T.
, Kengne
, J.
, Tapche
, R. W.
, and Pelap
, F. B.
, 2018
, “Uncertain Destination Dynamics of a Novel Memristive 4D Autonomous System
,” Chaos, Solitons Fractals
, 107
, pp. 177
–185
.38.
Liao
, T. L.
, 1998
, “Adaptive Synchronization of Two Lorenz Systems
,” Chaos, Solitons Fractals
, 9
(9
), pp. 1555
–1561
.39.
Lorenz
, E. N.
, 1963
, “Deterministic Nonperiodic Flow
,” J. Atmos. Sci.
, 20
(2
), pp. 130
–141
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