忆阻Hopfield神经网络的初值位移调控动力学及其图像加密应用

doi:10.19596/j.cnki.1001-246x.8547

摘要/Abstract

摘要：

利用改进的多稳态忆阻器模拟神经元耦合突触，提出一种忆阻Hopfield神经网络(HNN)模型。用分岔图、Lyapunov指数谱、相图、庞加莱截面等对其动力学行为进行理论分析与数值仿真。结果表明: 该忆阻HNN不仅能够产生不同拓扑结构的混沌吸引子，而且能够产生高度依赖忆阻初值的位移调控动力学行为。最后，基于该忆阻HNN设计一种混沌图像加密系统，重点分析系统的直方图、相关性、信息裔以及密钥敏感性。结果表明：所设计的图像加密算法能够有效抵抗各种内外统计分析攻击，具有较高的安全性。

关键词: 忆阻器, Hopfield神经网络, 初值位移调控, 混沌, 图像加密

Abstract:

A memristive Hopfield neural network (HNN) model is proposed in which an improved multi-stable memristor is used to simulate coupled neuron synapses. Dynamical behavior of the model is analyzed and simulated with bifurcation diagram, Lyapunov exponential spectrum, phase plot and Poincare section. It shows that the memristive HNN generates chaotic attractors with different topologies and generates initial offset boosting highly dependent on initial value of the memristor. Finally, a chaotic image encryption scheme is designed based on the memristive HNN. The histogram, correlation, information entropy and key sensitivity are analyzed. It shows that the image encryption scheme resists effectively various internal and external statistical analysis attacks and has higher security.

Key words: memristor, hopfield neural network, initial offset boosting, chaos, image encryption

孙亮, 罗佳, 乔印虎. 忆阻Hopfield神经网络的初值位移调控动力学及其图像加密应用[J]. 计算物理, 2023, 40(1): 106-116.

Liang SUN, Jia LUO, Yinhu QIAO. Initial Offset Boosting Dynamics in A Memristive Hopfield Neural Network and Its Application in Image Encryption[J]. Chinese Journal of Computational Physics, 2023, 40(1): 106-116.

图/表 15

图1 输入信号幅度/频率相关的忆阻器v-i曲线(a) F=0.2, 不同幅度下的紧磁滞回线; (b) A=2, 不同频率下的紧磁滞回线

Fig.1 Memristor v-i curve related to input signal amplitude/frequency (a) F = 0.2, pinched hysteresis loops at different amplitudes; (b) A=2, pinched hysteresis loops at different frequencies

图2 忆阻HNN拓扑图

Fig.2 Topology of memristive HNN

图3 函数f1(蓝色)与f2(红色)交点

Fig.3 Intersections of function f1 (blue) and function f2 (red)

表1 忆阻HNN的平衡点、特征值和稳定性

Table 1 Equilibrium points, eigenvalues, and stabilities of memristive HNN

线平衡点	特征值	平衡点稳定性
(0, 0, 0, 0, 0)	-1, 0.32±6.312j, 0.929 9±3.331 7j	不稳定鞍焦
(0, ±0.058 44, ±0.322 759, 0, ±3.12)	-1, 0.276 9±6.06j, 0.899 8±3.255 6j	不稳定鞍焦
(0, 0.043 198, 0.731 52, 0, ±6.24)	-1, -0.155 67±5.064 7j, 0.801 18±3.180 5j	不稳定鞍焦
(0, 0.067 25, 0.652 4, 0, ±9.36)	-1, 0.365 2±1.974 6j, 0.956 3±3.575 6j	不稳定鞍焦
(0, ±0.086 47, ±1.102 1, 0, ±12.48)	-1, 1.103 25±3.258 9j, 3.538, -5.59	不稳定鞍焦

图4 依赖参数μ的动力学行为(a) 分岔图; (b) 李雅普诺夫指数谱

Fig.4 Dynamical behavior related to parameter μ (a) bifurcation diagram; (b) Lyapunov exponents

图5 不同参数μ下，忆阻HNN的吸引子(a) 混沌吸引子μ=0.1，(b) 混沌吸引子μ=1，(c) 周期吸引子μ=1.255，(d) 混沌吸引子μ=1.5，(e) 准周期吸引子μ=1.8，(f) 混沌吸引子μ=1.89

Fig.5 Attractors of the memristive HNN under different parameter μ (a) chaotic attractor at μ=0.1, (b) chaotic attractor at μ=1, (c) periodic attractor at μ=1.255, (d) chaotic attractor at μ=1.5, (e) quasi-periodic attractor at μ=1.8, (f) chaotic attractor with μ=1.89

图6 关于初值φ0的分岔图

Fig.6 Bifurcation diagram about initial value φ0

图7 初值位移调控动力学(a) 共存6个吸引子相轨图，(b) 共存6个混沌序列

Fig.7 Initial offset boosting dynamics (a) coexisting six attractors, (b) coexisting six chaotic sequences

图8 庞加莱截面(a) x1=0; (b) x2=0; (c) x4=0

Fig.8 Poincaré section (a) x1=0; (b) x2=0; (c) x4=0

图9 吸引盆动力学图

Fig.9 Basin of attraction

图10 混沌图像加密系统

Fig.10 A chaotic image encryption system

图11 (a)~(c) 原始图像; (d)~(f) 原始图像的直方图; (g)~(i) 加密图像; (j)~(l)加密图像的直方图

Fig.11 (a)-(c) Plain images; (d)-(f) Histograms of the plain images; (g)-(i) Encrypted images; (j)-(l) Histograms of the encrypted images

表2 原始图像和加密图像的相关系数与信息裔值

Table 2 Correlation coefficients and information entropies of original images and encrypted images

图片	水平相关系数	垂直相关系数	对角相关系数	信息裔
Lena原始图	0.970 303	0.941 693	0.925 557	7.374 0
Lena加密图	- 0.001 030	0.001 724	0.002 602	7.997 4
Peppers原始图	0.971 015	0.965 136	0.943 933	7.570 3
Peppers加密图	0.001 820	- 0.000 987	0.002 412	7.997 7
Baboon原始图	0.822 077	0.859 746	0.770 734	7.257 7
Baboon加密图	- 0.002 623	- 0.001 277 5	- 0.005 794	7.997 4

图12 (b)、(e)、(h)正确解密图; (c)、(f)、(i) 错误解密图

Fig.12 (b), (e), (h) Correct decryption images; (c), (f), (i) Wrong decryption images

表3 NPCR和UACI测试结果

Table 3 Test results of NPCR and UACI

	NPCR(Lena/Peppers/Baboon)	UACI(Lena/Peppers/Baboon)
Ref.[22]	99.58/--/--	33.64/--/--
Ref.[23]	99.65/99.84/--	33.64/33.68/--
Ref.[24]	99.606 3/--/99.609 4	33.295 8/--/33.328 2
本文	99.694 8/99.827 5/99.689 2	33.651 8/33.680 2/33.692 1

参考文献 24

1	MA J , TANG J . A review for dynamics in neuron and neuronal network[J]. Nonlinear Dynamics, 2017, 89 (3): 1569- 1578. DOI
2	KORN H , FAURE P . Is there chaos in the brain? Ⅱ. Experimental evidence and related models[J]. Comptes Rendus Biologies, 2003, 326 (9): 787- 840. DOI
3	TANG L , HE Z , YAO Y . Magnetic induction HR neuron with hidden extreme multistability and its circuit implementation[J]. Chinese Journal of Computational Physics, 2022, 39 (5): 589- 597.
4	曲良辉, 都琳, 胡海威, 等. 电磁刺激对FHN神经元系统的调控作用[J]. 动力学与控制学报, 2020, 18 (1): 40- 48.
5	LUO J , SUN L , QIAO Y . Dynamical analysis and circuit implementation of a memristor synapse-coupled ring Hopfield neural network[J]. Chinese Journal of Computational Physics, 2022, 39 (1): 109- 117.
6	ZHOU Q , WEI D . Firing activity of memristive neural network under field coupling[J]. Chinese Journal of Computational Physics, 2020, 37 (6): 750- 756.
7	LIN H , WANG C , CHEN C , et al. Neural bursting and synchronization emulated by neural networks and circuits[J]. IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 2021, 68 (8): 3397- 3410. DOI
8	XU Q , SONG Z , QIAN H , et al. Numerical analyses and breadboard experiments of twin attractors in two-neuron-based non-autonomous Hopfield neural network[J]. The European Physical Journal Special Topics, 2018, 227 (7): 777- 786.
9	NJITACKE Z T , ISAAC S D , KENGNE J , et al. Extremely rich dynamics from hyperchaotic Hopfield neural network: Hysteretic dynamics, parallel bifurcation branches, coexistence of multiple stable states and its analog circuit implementation[J]. The European Physical Journal Special Topics, 2020, 229 (6): 1133- 1154.
10	NJITACKE Z T , KENGNE J , FOTSIN H B . Coexistence of multiple stable states and bursting oscillations in a 4D Hopfield neural network[J]. Circuits, Systems, and Signal Processing, 2020, 39 (7): 3424- 3444. DOI
11	王春华, 蔺海荣, 孙晶如, 等. 基于忆阻器的混沌、存储器及神经网络电路研究进展[J]. 电子与信息学报, 2020, 42 (4): 795- 810.
12	陈墨, 陈成杰, 包伯成, 等. 忆阻突触耦合Hopfield神经网络的初值敏感动力学[J]. 电子与信息学报, 2020, 42 (4): 870- 877.
13	蒋欣, 王光义, 周玮. 忆阻Hopfield神经网络的混沌特性研究[J]. 杭州电子科技大学学报(自然科学版), 2022, 42 (2): 1- 7.
14	TANG L , HE Z , YAO Y . Dynamical analysis and circuit implementation of a memristive Hopfield neural network[J]. Chinese Journal of Computational Physics, 2022, 39 (2): 244- 252.
15	BAO B , QIAN H , XU Q , et al. Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network[J]. Frontiers in Computational Neuroscience, 2017, 11, 81. DOI
16	LIN H , WANG C , HONG Q , et al. A multi-stable memristor and its application in a neural network[J]. IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 2020, 67 (12): 3472- 3476.
17	ZHANG S , ZHENG J , WANG X , et al. Initial offset boosting coexisting attractors in memristive multi-double-scroll Hopfield neural network[J]. Nonlinear Dynamics, 2020, 102 (4): 2821- 2841.
18	LAI Q , WAN Z , ZHANG H , et al. Design and analysis of multiscroll memristive Hopfield neural network with adjustable memductance and application to image encryption[J]. IEEE Transactions on Neural Networks and Learning Systems, DOI
19	CHEN M , XUE R , WU H , et al. Periodically varied initial offset boosting behaviors in a memristive system with cosine memductance[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20 (12): 1706- 1717.
20	WU H G , YE Y , BAO B C , et al. Memristor initial boosting behaviors in a two-memristor-based hyperchaotic system[J]. Chaos, Solitons & Fractals, 2019, 121, 178- 185.
21	YAN M , XU H . A chaotic system with adjustable number of coexisting attractors[J]. Chinese Journal of Computational Physics, 2021, 38 (2): 244- 252.
22	XUE W , ZHANG Y . Application of conservative hyperchaos in digital image encryption[J]. Chinese Journal of Computational Physics, 2020, 37 (4): 497- 504.
23	CHENG G , WANG C , XU C . A novel hyper-chaotic image encryption scheme based on quantum genetic algorithm and compressive sensing[J]. Multimedia Tools and Applications, 2020, 79 (39): 29243- 29263.
24	IQBAL N , HANIF M , REHMAN Z U , et al. On the novel image encryption based on chaotic system and DNA computing[J]. Multimedia Tools and Applications, 2022, 81 (6): 8107- 8137.

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[13]	赵明, 王柯, 余端民. 圆内开缝圆自然对流的Ruelle-Takens混沌道路[J]. 计算物理, 2020, 37(6): 667-676.
[14]	薛薇, 张永超. 保守超混沌在数字图像加密中的应用[J]. 计算物理, 2020, 37(4): 497-504.
[15]	石建平, 李培生, 刘国平. 基于改进克隆选择算法的统一混沌系统同步控制与参数辨识[J]. 计算物理, 2020, 37(2): 240-252.