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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Fig. 4

An improved switch-modulated digital secure communication system

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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

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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)).

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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)).

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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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