Multivariable Function Projective Synchronization of High Dimensional Chaotic Systems and Its Secure Communication Scheme

doi:10.19596/j.cnki.1001-246x.8550

Abstract

Abstract:

A modified function projective synchronization scheme, multivariable function projective synchronization, is proposed. Different from existing function projective synchronization schemes whose scaling factor is unary function, the scaling factor discussed in this scheme is multivariable function. It implies that synchronization behavior of the proposed scheme is more complicated and more unpredictable. The driver and responding system show correlation in a higher dimension. With modified active control method, controller of the synchronization is designed. In numerical simulations, four-dimensional energy resources system and a new hyperchaotic Chua system are investigated. Multivariable function projective synchronization with two types of scaling factor between two systems are achieved. Based on the synchronization, a secure communication scheme is designed. A secure communication simulation is carried out with sinusoidal signal and gray image signal as information signals. Meanwhile, anti-noise simulation is performed with a wavelet threshold denoising algorithm. It shows that the synchronization method is feasible, and the secure communication scheme is effective.

Key words: multivariable function projective synchronization, modified active control, secure communication, wavelet threshold denoising

Zhenbo LI, Yezhi TANG. Multivariable Function Projective Synchronization of High Dimensional Chaotic Systems and Its Secure Communication Scheme[J]. Chinese Journal of Computational Physics, 2023, 40(1): 91-105.

Figures/Tables 16

Table 1 Different kinds of projective synchronization

同步误差向量	同步类型	比例因子Γ
e_i=x_i-Γy_i 其中x_i为驱动向量，y_i为响应向量，Γ为比例因子	完全同步	Γ=1
	投影同步	Γ为常数，大小由驱动系统初值决定
	广义投影同步	Γ为常数，大小可任意指定
	一元函数投影同步	Γ为时间t的一元函数
	二元函数投影同步	Γ为二元函数
	多元函数投影同步	Γ为多元函数

Fig.1 Time response of System (8) and (9) with scaling function Γ1 (a) time response of x1 and x2; (b) time response of y1 and y2; (c) time response of z1 and z2; (d) time response of ω1 and ω2

Fig.2 Phase portraits of System (8) and (9) with scaling function Γ1 (a) phase portraits in x-y-z space; (b) phase portraits in y-z-ω space

Fig.3 Synchronization results of System (8) and (9) with scaling function Γ1 (a) synchronization results of x1 and x2; (b) synchronization results of y1 and y2; (c) synchronization results of z1 and z2; (d) synchronization results of ω1 and ω2

Fig.4 Synchronization errors of System (8) and (9) with scaling function Γ1

Fig.5 Synchronization errors of System (8) and (9) with scaling function Γ1 at different ki (a) ki=1; (b) ki=10

Fig.6 Synchronization errors of System (8) and (9) (a) nonlinear feedback method, (b) adaptive feedback method

Fig.7 Time response of System (8) and (9) with scaling function Γ2 (a) time response of x1 and x2; (b) time response of y1 and y2; (c) time response of z1 and z2; (d) time response of ω1 and ω2

Fig.8 Phase portraits of System (8) and (9) with scaling function Γ2 (a) phase portrait in x-y-z space; (b) phase portrait in y-z-ω space

Fig.9 Synchronization results of System (8) and (9) with scaling function Γ2 (a) synchronization results of and x2; (b) synchronization results of y1 and y2; (c) synchronization results of z1 and z2; (d) synchronization results of ω1 and ω2

Fig.10 Synchronization errors of System (8) and (9) with scaling function Γ2

Fig.11 A flow chart of wavelet threshold denoising

Fig.12 A flow chart of secure communication scheme

Fig.13 Pictures of communication signal θ(t) at different SNR

Fig.14 Recovered signal of sinusoidal signal at different SNR

Fig.15 Recovered image of grayscale image at different SNR

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