高维混沌系统的多元函数投影同步及其保密通信方案

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

摘要/Abstract

摘要：

提出一种改进的函数投影同步方案——多元函数投影同步。与现有的比例因子为一元函数的函数投影同步方案不同，本文讨论的比例因子为多元函数。在该方案下，系统间的同步行为更复杂、更难以预测，驱动与响应系统将在更高的维度中展现出相关性。基于改进的主动控制方法，研究四维能源系统和超混沌蔡氏系统的多元函数投影同步问题，并设计同步控制器。在数值模拟中，分别针对比例因子为多元线性函数和多元非线性函数情况进行同步仿真，实现了系统之间的多元函数投影同步。利用上述同步，设计相应的保密通信方案，以正弦信号和灰度图像信号作为信息信号进行保密通信仿真，并通过引入小波阈值去噪算法进行抗噪声模拟。结果表明: 提出的同步方案可行，所设计的保密通信方法有效，具备较好的保密性和稳健性。

关键词: 多元函数投影同步, 改进的主动控制, 保密通信, 小波阈值去噪

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

李震波, 唐叶芝. 高维混沌系统的多元函数投影同步及其保密通信方案[J]. 计算物理, 2023, 40(1): 91-105.

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.

图/表 16

表1 各类常见的投影同步

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

图1 系统(8)和系统(9)在比例因子Γ1下同步的时间历程(a) x1与x2的时间历程；(b) y1与y2的时间历程；(c) z1与z2的时间历程；(d) ω1与ω2的时间历程

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

图2 系统(8)和系统(9)在比例因子Γ1下同步的相图(a) x-y-z空间的相图；(b) y-z-ω空间的相图

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

图3 系统(8)和系统(9)关于比例因子Γ1下的同步结果(a) x1与x2的同步结果; (b) y1与y2的同步结果；(c) z1与z2的同步结果；(d) ω1与ω2的同步结果

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

图4 系统(8)和系统(9)在比例因子Γ1下的同步误差曲线

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

图5 系统(8)和系统(9)在比例因子Γ1下的同步误差曲线(a) ki=1；(b) ki=10

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

图6 系统(8)和系统(9)的同步误差曲线(a) 非线性反馈控制法；(b) 自适应反馈控制法

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

图7 系统(8)和系统(9)在比例因子Γ2下同步的时间历程(a) x1与x2的时间历程；(b) y1与y2的时间历程；(c) z1与z2的时间历程；(d) ω1与ω2的时间历程

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

图8 系统(8)和系统(9)在比例因子Γ2下同步的相图(a) x-y-z空间的相图；(b) y-z-ω空间的相图

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

图9 系统(8)和系统(9)在比例因子Γ2下的同步结果(a) x1与x2的同步结果；(b) y1与y2的同步结果；(c) z1与z2的同步结果; (d) ω1与ω2的同步结果

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

图10 系统(8)和系统(9)在比例因子Γ2下的同步误差曲线

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

图11 小波阈值去噪原理

Fig.11 A flow chart of wavelet threshold denoising

图12 保密通信原理图

Fig.12 A flow chart of secure communication scheme

图13 通信信号θ(t)在不同信噪比下的图像

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

图14 不同信噪比下正弦信号的恢复信号

Fig.14 Recovered signal of sinusoidal signal at different SNR

图15 不同信噪比下灰度图像的恢复信号

Fig.15 Recovered image of grayscale image at different SNR

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