Research Papers

Hybrid Delayed Synchronizations of Complex Chaotic Systems in Modulus-Phase Spaces and Its Application

[+] Author and Article Information
Luo Chao

School of Information Science and Engineering,
Shandong Normal University,
Jinan 250014, China;
Shandong Provincial Key Laboratory
for Novel Distributed Computer Software Technology,
Jinan 250014, China
e-mail: cluo79@gmail.com

1Corresponding author.

Manuscript received April 21, 2015; final manuscript received November 3, 2015; published online December 10, 2015. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 11(4), 041010 (Dec 10, 2015) (8 pages) Paper No: CND-15-1102; doi: 10.1115/1.4031956 History: Received April 21, 2015; Revised November 03, 2015

Compared with chaotic systems over the real number field, complex chaotic dynamics have some unique properties. In this paper, a kind of novel hybrid synchronizations of complex chaotic systems is discussed analytically and numerically. Between two nonidentical complex chaotic systems, modified projective synchronization (MPS) in the modulus space and complete synchronization in the phase space are simultaneously achieved by means of active control. Based on the Lyapunov stability theory, a controller is developed, in which time delay as an important consideration is involved. Furthermore, a switch-modulated digital secure communication system based on the proposed synchronization scheme is carried out. Different from the previous works, only one set of drive-response chaotic systems can implement switch-modulated secure communication, which could simplify the complexity of design. Furthermore, the latency of a signal transmitted between transmitter and receiver is simulated by channel delay. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.

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Lang, R. L. , 2010, “ A Stochastic Complex Model With Random Imaginary Noise,” Nonlinear Dyn., 62(3), pp. 561–565. [CrossRef]
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. [CrossRef] [PubMed]
Vladimirov, A. G. , Toronov, V. Y. , and Derbov, V. L. , 1998, “ Properties of the Phase Space and Bifurcations in the Complex Lorenz Model,” Tech. Phys., 43(8), pp. 877–884. [CrossRef]
Xu, Y. , Xu, W. , and Mahmoud, G. M. , 2004, “ On a Complex Beam–Beam Interaction Model With Random Forcing,” Phys. A, 336(3), pp. 347–360. [CrossRef]
Xu, Y. , Mahmoud, G. M. , Xu, W. , and Lei, Y. , 2005, “ Suppressing Chaos of a Complex Duffing's System Using a Random Phase,” Chaos, Solitons Fractals, 23(1), pp. 265–273. [CrossRef]
Mahmoud, G. M. , and Aly, S. A. , 2000, “ Periodic Attractors of Complex Damped Non-Linear Systems,” Int. J. Nonlinear Mech., 35(2), pp. 309–323. [CrossRef]
Mahmoud, G. M. , Mahmoud, E. E. , and Arafa, A. A. , 2013, “ On Projective Synchronization of Hyperchaotic Complex Nonlinear Systems Based on Passive Theory for Secure Communications,” Phys. Scr., 87(5), p. 055002. [CrossRef]
Fowler, A. C. , Gibbon, J. D. , and McGuinness, M. J. , 1982, “ The Complex Lorenz Equations,” Phys. D, 4(2), pp. 139–163. [CrossRef]
Mahmoud, G. M. , Al-Kashif, M. A. , and Aly, S. A. , 2007, “ Basic Properties and Chaotic Synchronization of Complex Lorenz System,” Int. J. Mod. Phys. C, 18(2), pp. 253–265. [CrossRef]
Mahmoud, G. M. , Mahmoud, E. E. , and Ahmed, M. E. , 2009, “ On the Hyperchaotic Complex Lü System,” Nonlinear Dyn., 58(4), pp. 725–738. [CrossRef]
Mahmoud, G. M. , Mahmoud, E. E. , and Arafa, A. A. , 2015, “ On Modified Time Delay Hyperchaotic Complex Lü System,” Nonlinear Dyn., 80(1–2), pp. 855–869. [CrossRef]
Mahmoud, G. M. , Ahmed, M. E. , and Mahmoud, E. E. , 2008, “ Analysis of Hyperchaotic Complex Lorenz Systems,” Int. J. Mod. Phys. C, 19(10), pp. 1477–1494. [CrossRef]
Mahmoud, E. E. , 2012, “ Dynamics and Synchronization of New Hyperchaotic Complex Lorenz System,” Math. Comput. Modell., 55(7), pp. 1951–1962. [CrossRef]
Luo, C. , and Wang, X. , 2013, “ Chaos in the Fractional-Order Complex Lorenz System and Its Synchronization,” Nonlinear Dyn., 71(1–2), pp. 241–257. [CrossRef]
Peng, J. H. , Ding, E. J. , Ding, M. , and Yang, W. , 1996, “ Synchronizing Hyperchaos With a Scalar Transmitted Signal,” Phys. Rev. Lett., 76(6), pp. 904–907. [CrossRef] [PubMed]
Liu, P. , Song, H. , and Li, X. , 2015, “ Observe-Based Projective Synchronization of Chaotic Complex Modified Van der Pol-Duffing Oscillator With Application to Secure Communication,” ASME J. Comput. Nonlinear Dyn., 10(5), p. 051015. [CrossRef]
Li, C. L. , 2012, “ Tracking Control and Generalized Projective Synchronization of a Class of Hyperchaotic System With Unknown Parameter and Disturbance,” Commun. Nonlinear Sci. Numer. Simul., 17(1), pp. 405–413. [CrossRef]
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. [CrossRef]
Zhang, F. , and Liu, S. , 2014, “ Full State Hybrid Projective Synchronization and Parameters Identification for Uncertain Chaotic (Hyperchaotic) Complex Systems,” ASME J. Comput. Nonlinear Dyn., 9(2), p. 021009. [CrossRef]
Chen, D. Y. , Zhang, R. F. , Sprott, J. C. , Chen, H. T. , and Ma, X. Y. , 2012, “ Synchronization Between Integer-Order Chaotic Systems and a Class of Fractional-Order Chaotic Systems Via Sliding Mode Control,” Chaos, 22(2), p. 023109. [CrossRef] [PubMed]
Yang, J. , Chen, Y. , and Zhu, F. , 2014, “ Singular Reduced-Order Observer-Based Synchronization for Uncertain Chaotic Systems Subject to Channel Disturbance and Chaos-Based Secure Communication,” Appl. Math. Comput., 229(6), pp. 227–238. [CrossRef]
Pai, M. C. , 2014, “ Global Synchronization of Uncertain Chaotic Systems Via Discrete-Time Sliding Mode Control,” Appl. Math. Comput., 227(2), pp. 663–671. [CrossRef]
Liu, S. T. , and Liu, P. , 2011, “ Adaptive Anti-Synchronization of Chaotic Complex Nonlinear Systems With Unknown Parameters,” Nonlinear Anal.: Real World Appl., 12(6), pp. 3046–3055. [CrossRef]
Liu, P. , Liu, S. T. , and Li, X. , 2012, “ Adaptive Modified Function Projective Synchronization of General Uncertain Chaotic Complex Systems,” Phys. Scr., 85(3), p. 035005. [CrossRef]
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. [CrossRef]
Nian, F. Z. , and Wang, X. Y. , 2010, “ Module-Phase Synchronization in Complex Dynamic System,” Appl. Math. Comput., 217(6), pp. 2481–2489. [CrossRef]
Wang, X. Y. , and Luo, C. , 2013, “ Hybrid Modulus-Phase Synchronization of Hyperchaotic Complex Systems and Its Application to Secure Communication,” Int. J. Nonlinear Sci. Numer. Simul., 14(7–8), pp. 533–542.
Pourdehi, S. , Karimaghaee, P. , and Karimipour, D. , 2011, “ Adaptive Controller Design for Lag-Synchronization of Two Non-Identical Time-Delayed Chaotic Systems With Unknown Parameters,” Phys. Lett. A, 375(17), pp. 1769–1778. [CrossRef]
Mahmoud, G. M. , Mahmoud, E. E. , and Ahmed, M. E. , 2007, “ A Hyperchaotic Complex Chen System and Its Dynamics,” Int. J. Appl. Math. Stat., 12(D07), pp. 90–100.
Yang, J. , and Zhu, F. , 2013, “ Synchronization for Chaotic Systems and Chaos-Based Secure Communications Via Both Reduced-Order and Step-by-Step Sliding Mode Observers,” Commun. Nonlinear Sci. Numer. Simul., 18(4), pp. 926–937. [CrossRef]
Pan, J. , Ding, Q. , and Du, B. X. , 2012, “ A New Improved Scheme of Chaotic Masking Secure Communication Based on Lorenz System,” Int. J. Bifurcation Chaos, 22(5), p. 1250125. [CrossRef]
Sheikhan, M. , Shahnazi, R. , and Garoucy, S. , 2013, “ Synchronization of General Chaotic Systems Using Neural Controllers With Application to Secure Communication,” Neural Comput. Appl., 22(2), pp. 361–373. [CrossRef]


Grahic Jump Location
Fig. 1

Three-dimensional projections of drive and response hyperchaotic complex systems: (a) system (15) with (a1,a2,a3,a4)T=(14,39,5,12)T and (b) system (16) with (b1,b2,b3,b4)T=(32,25,4,10)T

Grahic Jump Location
Fig. 2

The time evolution of synchronization errors between systems (15) and (16). MPS in modulus space ((a) and (b)) and complete synchronization in phase space ((c) and (d)).

Grahic Jump Location
Fig. 3

Modulus and phase values of state variables of systems (15) and (16). MPS in modulus space ((a) and (b)) and complete synchronization in phase space ((c) and (d)).

Grahic Jump Location
Fig. 4

An improved switch-modulated digital secure communication system

Grahic Jump Location
Fig. 5

Simulation results of improved switch-modulated digital secure communication system: (a) the original digital signal s(t), (b) the transmitted signal s̃(t), (c) the recovered signal s′(t + τ), (d) s(t)−s′(t + τ), (e) s̃(t)−M(y1(t + τ))/c1, (f) s̃(t)−‖α′(y1(t + τ))P(y1(t + τ))‖, (g) partial enlargement of Fig. 5(e), and (h) partial enlargement of Fig. 5(f)




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