Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.
Issue Section:
Research Papers
Keywords:
Stability
References
1.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, New York.
2.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, Theory and Application of Fractional Differential Equations
, Elsevier
, New York
.3.
Koeller
, R. C.
, 1984
, “Applications of Fractional Calculus to the Theory of Viscoelasticity
,” ASME J. Appl. Mech.
, 51
(2
), pp. 299
–307
.4.
Sabatier
, J.
, Agrawal
, O. P.
, and Machado
, J. T.
, 2007
, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
, Springer
, Dordrecht, The Netherlands
.5.
Machado
, J. T.
, Kiryakova
, V.
, and Mainardi
, F.
, 2011
, “Recent History of Fractional Calculus
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
(3
), pp. 1140
–1153
.6.
Hartley
, T. T.
, Lorenzo
, C. F.
, and Killory Qammer
, H.
, 1995
, “Chaos in a Fractional Order Chua's System
,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.
, 42
(8
), pp. 485
–490
.7.
Razminia
, A.
, and Baleanu
, D.
, 2013
, “Fractional Hyperchaotic Telecommunication Systems: A New Paradigm
,” ASME J. Comput. Nonlinear Dyn.
, 8
(3
), p. 031012
.8.
Wang
, X. Y.
, and Zhang
, H.
, 2013
, “Bivariate Module-Phase Synchronization of a Fractional-Order Lorenz System in Different Dimensions
,” ASME J. Comput. Nonlinear Dyn.
, 8
(3
), p. 031017
.9.
Li
, C.
, and Chen
, G.
, 2004
, “Chaos and Hyperchaos in the Fractional-Order Rossler Equations
,” Phys. A
, 341
, pp. 55
–61
.10.
Grigorenko
, I.
, and Grigorenko
, E.
, 2003
, “Chaotic Dynamics of the Fractional Lorenz System
,” Phys. Rev. Lett.
, 91
(3
), p. 034101
.11.
Li
, C.
, and Chen
, G.
, 2004
, “Chaos in the Fractional Order Chen System and Its Control
,” Chaos, Solitons Fract.
, 22
(3
), pp. 549
–554
.12.
Lu
, J. G.
, 2006
, “Chaotic Dynamics of the Fractional-Order Lu System and Its Synchronization
,” Phys. Lett. A
, 354
(4
), pp. 305
–311
.13.
Gibbon
, J. D.
, and McGuinness
, M. J.
, 1982
, “The Real and Complex Lorenz Equations in Rotating Fluids and Lasers
,” Phys. D
, 5
(1
), pp. 108
–122
.14.
Fowler
, A. C.
, Gibbon
, J. D.
, and McGuinness
, M. J.
, 1982
, “The Complex Lorenz Equations
,” Phys. D
, 4
(2
), pp. 139
–163
.15.
Fowler
, A. C.
, Gibbon
, J. D.
, and McGuinness
, M. J.
, 1983
, “The Real and Complex Lorenz Equations and Their Relevance to Physical Systems
,” Phys. D
, 7
(1
), pp. 126
–134
.16.
Ning
, C. Z.
, and Haken
, H.
, 1990
, “Detuned Lasers and the Complex Lorenz Equations: Subcritical and Supercritical Hopf Bifurcations
,” Phys. Rev. A
, 41
(7
), pp. 3826
–3837
.17.
Li
, C.
, Liao
, X.
, and Yu
, J.
, 2003
, “Synchronization of Fractional Order Chaotic Systems
,” Phys. Rev. E
, 68
(6
), p. 067203
.18.
Zhou
, P.
, and Zhu
, W.
, 2011
, “Function Projective Synchronization for Fractional-Order Chaotic Systems
,” Nonlinear Anal.: Real World Appl.
, 12
(2
), pp. 811
–816
.19.
Bhalekar
, S.
, and Daftardar-Gejji
, V.
, 2010
, “Synchronization of Different Fractional Order Chaotic Systems Using Active Control
,” Commun. Nonlinear Sci. Numer. Simul.
, 15
(11
), pp. 3536
–3546
.20.
Aghababa
, M. P.
, 2014
, “Synchronization and Stabilization of Fractional Second-Order Nonlinear Complex Systems
,” Nonlinear Dyn.
, 80
(4
), pp. 1731
–1744
.21.
Sun
, J.
, and Shen
, Y.
, 2014
, “Adaptive Anti-Synchronization of Chaotic Complex Systems and Chaotic Real Systems With Unknown Parameters
,” J. Vib. Control
(in press).22.
Ma
, T.
, and Zhang
, J.
, 2015
, “Hybrid Synchronization of Coupled Fractional-Order Complex Networks
,” Neurocomputing
, 157
, pp. 166
–172
.23.
Mahmoud
, G. M.
, and Mahmoud
, E. E.
, 2010
, “Synchronization and Control of Hyperchaotic Complex Lorenz System
,” Math. Comput. Simul.
, 80
(12
), pp. 2286
–2296
.24.
Mahmoud
, G. M.
, Mahmoud
, E. E.
, Farghaly
, A. A.
, and Aly
, S. A.
, 2009
, “Chaotic Synchronization of Two Complex Nonlinear Oscillators
,” Chaos, Solitons Fract.
, 42
(5
), pp. 2858
–2864
.25.
Mahmoud
, G. M.
, Aly
, S. A.
, and Al-Kashif
, M. A.
, 2008
, “Dynamical Properties and Chaos Synchronization of a New Chaotic Complex Nonlinear System
,” Nonlinear Dyn.
, 51
(1–2
), pp. 171
–181
.26.
Wang
, W. X.
, Huang
, L.
, Lai
, Y. C.
, and Chen
, G.
, 2009
, “Onset of Synchronization in Weighted Scale-Free Networks
,” Chaos
, 19
(1
), p. 013134
.27.
Xie
, Y.
, Gong
, Y.
, Hao
, Y.
, and Ma
, X.
, 2010
, “Synchronization Transitions on Complex Thermo-Sensitive Neuron Networks With Time Delays
,” Biophys. chem.
, 146
(2
), pp. 126
–132
.28.
Yang
, Y.
, Duan
, X.
, and Li
, L.
, 2011
, “Wireless Sensor Network Time Synchronization Design for Large Generator On-Line Monitoring
,” Sensor Lett.
, 9
(4
), pp. 1467
–1471
.29.
Mahmoud
, G. M.
, and Mahmoud
, E. E.
, 2010
, “Complete Synchronization of Chaotic Complex Nonlinear Systems With Uncertain Parameters
,” Nonlinear Dyn.
, 62
(4
), pp. 875
–882
.30.
Chen
, D.
, Zhao
, W.
, Liu
, X.
, and Ma
, X.
, 2015
, “Synchronization and Antisynchronization of a Class of Chaotic Systems With Nonidentical Orders and Uncertain Parameters
,” ASME J. Comput. Nonlinear Dyn.
, 10
(1
), p. 011003
.31.
Kim
, C. M.
, Rim
, S.
, Kye
, W. H.
, Ryu
, J. W.
, and Park
, Y. J.
, 2003
, “Anti-Synchronization of Chaotic Oscillators
,” Phys. Lett. A
, 320
(1
), pp. 39
–46
.32.
Kocarev
, L.
, and Parlitz
, U.
, 1996
, “Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems
,” Phys. Rev. Lett.
, 76
(11
), pp. 1816
–1819
.33.
Mainieri
, R.
, and Rehacek
, J.
, 1999
, “Projective Synchronization in Three-Dimensional Chaotic Systems
,” Phys. Rev. Lett.
, 82
(15
), pp. 3042
–3045
.34.
Wang
, S.
, Yu
, Y.
, and Wen
, G.
, 2014
, “Hybrid Projective Synchronization of Time-Delayed Fractional Order Chaotic Systems
,” Nonlinear Anal. Hybrid Syst.
, 11
, pp. 129
–138
.35.
Hu
, M.
, Yang
, Y.
, Xu
, Z.
, and Guo
, L.
, 2008
, “Hybrid Projective Synchronization in a Chaotic Complex Nonlinear System
,” Math. Comput. Simul.
, 79
(3
), pp. 449
–457
.36.
Liu
, P.
, 2015
, “Adaptive Hybrid Function Projective Synchronization of General Chaotic Complex Systems With Different Orders
,” ASME J. Comput. Nonlinear Dyn.
, 10
(2
), p. 021018
.37.
Sun
, J.
, Shen
, Y.
, and Zhang
, X.
, 2014
, “Modified Projective and Modified Function Projective Synchronization of a Class of Real Nonlinear Systems and a Class of Complex Nonlinear Systems
,” Nonlinear Dyn.
, 78
(3
), pp. 1755
–1764
.38.
Liu
, J.
, Liu
, S.
, and Yuan
, C.
, 2014
, “Adaptive Complex Modified Projective Synchronization of Complex Chaotic (Hyperchaotic) Systems With Uncertain Complex Parameters
,” Nonlinear Dyn.
, 79
(2
), pp. 1035
–1047
.39.
Mahmoud
, G. M.
, and Mahmoud
, E. E.
, 2010
, “Phase and Antiphase Synchronization of Two Identical Hyperchaotic Complex Nonlinear Systems
,” Nonlinear Dyn.
, 61
(1–2
), pp. 141
–152
.40.
Chee
, C. Y.
, and Xu
, D.
, 2006
, “Chaos-Based M-Nary Digital Communication Technique Using Controller Projective Synchronization
,” IEE Proc-G Circ. Dev. Syst.
, 153
(4
), pp. 357
–360
.41.
Asl
, F. M.
, and Ulsoy
, A. G.
, 2003
, “Analysis of a System of Linear Delay Differential Equations
,” ASME J. Dyn. Syst., Meas., Control
, 125
(2
), pp. 215
–223
.42.
Chen
, Y.
, and Moore
, K. L.
, 2002
, “Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems
,” Nonlinear Dyn.
, 29
(1
), pp. 191
–200
.43.
Deng
, W.
, Li
, C.
, and Lu
, J.
, 2007
, “Stability Analysis of Linear Fractional Differential System With Multiple Time Delays
,” Nonlinear Dyn.
, 48
(4
), pp. 409
–416
.44.
Li
, L.
, Peng
, H.
, Yang
, Y.
, and Wang
, X.
, 2009
, “On the Chaotic Synchronization of Lorenz Systems With Time-Varying Lags
,” Chaos, Solitons Fract.
, 41
(2
), pp. 783
–794
.45.
Zhang
, F.
, 2015
, “Lag Synchronization of Complex Lorenz System With Applications to Communication
,” Entropy
, 17
(7
), pp. 4974
–4985
.46.
Luo
, C.
, and Wang
, X.
, 2013
, “Chaos Generated From the Fractional-Order Complex Chen System and Its Application to Digital Secure Communication
,” Int. J. Mod. Phys. C
, 24
(4
), p. 1350025
.47.
Mahmoud
, G. M.
, Mahmoud
, E. E.
, and Arafa
, A. A.
, 2015
, “On Modified Time Delay Hyperchaotic Complex Lu System
,” Nonlinear Dyn.
, 80
(1–2
), pp. 855
–869
.48.
Diethelm
, K.
, Ford
, N. J.
, and Freed
, A. D.
, 2002
, “A Predictor–Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlinear Dyn.
, 29
(1–4
), pp. 3
–22
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