可控多双涡卷忆阻Hopfield神经网络建模及其动力学分析

刘嵩; 李子涵; 邱达; 罗敏; 赖强

doi:10.11999/JEIT250972

可控多双涡卷忆阻Hopfield神经网络建模及其动力学分析

doi: 10.11999/JEIT250972 cstr: 32379.14.JEIT250972

刘嵩^1, ,,
李子涵¹,
邱达¹,
罗敏¹,
赖强²

1.
湖北民族大学智能科学与工程学院恩施 445000
2.
华东交通大学电气与自动化工程学院南昌 330013

基金项目: 湖北省自然科学基金 (2024AFD068)，中国高校产学研创新基金(2024IT115)

详细信息

作者简介:
刘嵩：男，副教授，研究方向为混沌理论，忆阻神经网络，信息安全，智能控制

李子涵：男，硕士生，研究方向为忆阻神经网络

邱达：男，工程师，研究方向为智能检测与控制，光纤光栅传感，滑模切换系统，忆阻混沌系统

罗敏：女，副教授，研究方向为混沌理论，忆阻电路，模式识别，光电信号处理

赖强：男，教授，研究方向为混沌理论与应用、信息安全、忆阻电路、神经网络、多智能体系统、人工智能

通讯作者:
刘嵩　liusong@hbmzu.edu.cn

中图分类号: TM132; TN711.4
计量
- 文章访问数: 26
- HTML全文浏览量: 8
- PDF下载量: 2
- 被引次数: 0
出版历程
- 修回日期: 2025-12-01
- 录用日期: 2025-12-01
- 网络出版日期: 2025-12-09

Modeling and Dynamic Analysis of Controllable Multi-Double Scroll Memristor Hopfield Neural Network

LIU Song^{1
, ,},
LI Zihan¹,
QIU Da¹,
LUO Min¹,
LAI Qiang²

1.
School of College of Intelligent Systems Science and Engineering, Hubei Minzu University, Enshi 445000, China
2.
School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China

Funds: Natural Science Foundation of Hubei Province of China (2024AFD068), the Fund of China University Research Innovation (2024IT115)

摘要

摘要: 忆阻Hopfield神经网络是一种类脑神经网络，能够产生丰富的动力学行为。该文提出了一种新型包含反正切函数序列的忆阻器，将忆阻器耦合至神经网络中，可构建出一类包含电磁辐射与忆阻突触权重的忆阻全连接Hopfield神经网络。理论分析和数值仿真结果均表明，该模型可在相空间内生成单向、双向和三向多双涡旋混沌吸引子。进一步研究还发现，通过改变初始条件，发现该模型存在多个具有初始偏移增强特征的多双涡卷混沌吸引子，它们形状相同但位置不同，并且吸引子的数量以及双涡卷的个数均可控。此外改变忆阻突触耦合强度，结合分岔图和Lyapunov指数谱，发现该系统还存在丰富的共存对称吸引子，包括对称的周期吸引子与单涡卷混沌吸引子。最后基于FPGA平台完成了该系统的硬件实现，验证了该系统的物理存在性与可行性。
- 忆阻Hopfield神经网络 /
- 多双涡卷混沌吸引子 /
- 初始偏移增强 /
- FPGA硬件实现
Abstract: Objective:The human brain is a complex neural system capable of integrated information storage, computation, and parallel processing. The collective activity of neuronal populations processes and coordinates sensory inputs, resulting in highly nonlinear dynamics. Therefore, developing artificial neural network models and analyzing them with nonlinear dynamics theories is of considerable scientific and practical importance. As a brain-inspired model, the Hopfield neural network (HNN) can exhibit more diverse dynamics when a memristor is incorporated into its structure. Among various memristive neural networks, those generating multi-scroll attractors are particularly advantageous due to their richer dynamical behaviors and more complex topological structures, granting them significant research value and application potential in fields such as image encryption.Methods: A memristor model based on an arctangent function series is proposed and introduced into a fully connected HNN. This constructs a class of memristive HNNs that incorporate electromagnetic radiation effects and memristive synaptic weights. The generation mechanism of multi-double-scroll chaotic attractors is verified through equilibrium point analysis. Dynamical characteristics, including the effects of memristive synaptic coupling strength and initial offset-boosting behavior, are investigated using bifurcation diagrams, Lyapunov exponent spectra, and attraction basins. Finally, the system is implemented on an FPGA platform.Results and Discussions:This class of memristive HNN is capable of generating an arbitrary number of multi-directional multi-double-scroll chaotic attractors (Figs. 4, 5, 6). By adjusting the memristive synaptic coupling strength, various distinct types of coexisting attractors are observed (Figs. 7, 8). Furthermore, multiple coexisting multi-double-scroll chaotic attractors are revealed through modifications of the initial values (Figs. 9, 10, 11, 12). Finally, the hardware implementation on an FPGA (Figs. 13, 14) verifies the correctness and feasibility of the system.Conclusions:The findings indicate that the proposed model can generate unidirectional, bidirectional, and tridirectional multi-double-scroll chaotic attractors in phase space. The number of scrolls is precisely adjustable via the memristor’s control parameter. The system also exhibits initial-offset-boosting behavior, where the number of coexisting attractors is likewise regulated by this parameter. A higher-dimensional network can be constructed by increasing the number of memristive synapses, demonstrating the considerable universality of the proposed system. Benefiting from its intricate topology and rich dynamics, this network holds promising application prospects in engineering fields.
- Memristive neural network /
- Multi-Double-Scroll attractor /
- Initial offset boosting /
- FPGA hardware implementation

HTML全文

图 1 忆阻器特征

下载: 全尺寸图片幻灯片

图 2 忆阻Hopfield神经网络拓扑结构

下载: 全尺寸图片幻灯片

图 3 局部曲线图与吸引子相图

下载: 全尺寸图片幻灯片

图 4 模型1基于$ {\varphi }_{1} $方向可调控的多双涡卷混沌吸引子相图以及分岔图

下载: 全尺寸图片幻灯片

图 5 模型2网格多双涡卷混沌吸引子

下载: 全尺寸图片幻灯片

图 6 模型3网格多双涡卷混沌吸引子

下载: 全尺寸图片幻灯片

图 7 模型2随忆阻强度变化的动力学行为

下载: 全尺寸图片幻灯片

图 8 模型2共存对称吸引子

下载: 全尺寸图片幻灯片

图 9 模型1随初值$ {\varphi }_{1}(0) $变化的动力学行为

下载: 全尺寸图片幻灯片

图 10 模型2随初值$ {\varphi }_{2}(0) $变化的动力学行为

下载: 全尺寸图片幻灯片

图 11 模型2随初值$ {\varphi }_{1}(0) $变化的动力学行为

下载: 全尺寸图片幻灯片

图 12 模型3随初值$ {\varphi }_{3}(0) $变化的动力学行为

下载: 全尺寸图片幻灯片

图 13 硬件实现平台

下载: 全尺寸图片幻灯片

图 14 FPGA硬件实现结果

下载: 全尺寸图片幻灯片

表 1 忆阻神经网络模型与参数

编号	模型	固定参数	拓扑图
—	$ \begin{cases} {\dot{x}}_{1}=-{x}_{1}+2.22\tanh ({x}_{1})-1.21\tanh ({x}_{2})+0.5\tanh ({x}_{3})\\ {\dot{x}}_{2}=-{x}_{2}+2.1\tanh ({x}_{1})+1.7\tanh ({x}_{2})+1.15\tanh ({x}_{3})\\ {\dot{x}}_{3}=-{x}_{3}-4.75\tanh ({x}_{1})+0.1\tanh ({x}_{2})-1.135\tanh ({x}_{3})\\ \end{cases} $	—	图2(a)
1	$ \begin{cases} {\dot{x}}_{1}=-{x}_{1}+2.22\tanh ({x}_{1})-1.21\tanh ({x}_{2})+0.5\tanh ({x}_{3})\\ {\dot{x}}_{2}=-{x}_{2}+2.1\tanh ({x}_{1})+1.7\tanh ({x}_{2})+1.15\tanh ({x}_{3})\\ {\dot{x}}_{3}=-{x}_{3}-4.75\tanh ({x}_{1})+0.1\tanh ({x}_{2})+{W}_{1}({\varphi }_{1})\tanh ({x}_{3})\\ {\dot{\varphi }}_{1}=-{b}_{1}f({\varphi }_{1})+{\mathrm{a}}_{1}\tanh ({x}_{3})\\ \end{cases} $	$ {b}_{1}=1.39 $ $ {\alpha }_{1}=1 $ $ {\beta }_{1}=0.001 $	图2(b)
2	$ \begin{cases} {\dot{x}}_{1}=-{x}_{1}+2.22\tanh ({x}_{1})-1.21\tanh ({x}_{2})+0.5\tanh ({x}_{3})\\ {\dot{x}}_{2}=-{x}_{2}+2.1\tanh ({x}_{1})+1.7\tanh ({x}_{2})+1.15\tanh ({x}_{3})\\ {\dot{x}}_{3}=-{x}_{3}-4.75\tanh ({x}_{1})+{W}_{2}({\varphi }_{2})\tanh ({x}_{2})+{W}_{1}({\varphi }_{1})\tanh ({x}_{3})\\ {\dot{\varphi }}_{1}=-{b}_{1}f({\varphi }_{1})+{\mathrm{a}}_{1}\tanh ({x}_{3})\\ {\dot{\varphi }}_{2}=-{b}_{2}f({\varphi }_{2})+{\mathrm{a}}_{2}\tanh ({x}_{2})\\ \end{cases} $	$ {b}_{2}=1.385 $ $ {\alpha }_{2}=1 $ $ {\beta }_{2}=0.001 $ 其余参数与上一致	图2(c)
3	$ \begin{cases} {\dot{x}}_{1}=-{x}_{1}+2.22\tanh ({x}_{1})-1.21\tanh ({x}_{2})+0.5\tanh ({x}_{3})+{W}_{3}({\varphi }_{3}){x}_{1}\\ {\dot{x}}_{2}=-{x}_{2}+2.1\tanh ({x}_{1})+1.7\tanh ({x}_{2})+1.15\tanh ({x}_{3})\\ {\dot{x}}_{3}=-{x}_{3}-4.75\tanh ({x}_{1})+{W}_{2}({\varphi }_{2})\tanh ({x}_{2})+{W}_{1}({\varphi }_{1})\tanh ({x}_{3})\\ {\dot{\varphi }}_{1}=-{b}_{1}f({\varphi }_{1})+{\mathrm{a}}_{1}\tanh ({x}_{3})\\ {\dot{\varphi }}_{2}=-{b}_{2}f({\varphi }_{2})+{\mathrm{a}}_{2}\tanh ({x}_{2})\\ {\dot{\varphi }}_{3}=-{b}_{3}f({\varphi }_{3})+{a}_{3}{x}_{1}\\ \end{cases} $	$ {b}_{3}=1.4 $ $ {b}_{3}=1.4 $ $ {\beta }_{3}=0.001 $ 其余参数与上一致	图2(d)

下载: 导出CSV

表 2 模型2平衡点，特征值及平衡点类型

平衡点	特征值	平衡点类型
(0,0,0,0,0) (0,0,0,0,±2) (0,0,0,±2,0) (0,0,0,±2,±2)	–1.390, –1.385, 0.513, –0.364±1.340i	指标-1鞍焦点
(–0.422, –0.066, 1.016, 0.870, –0.142) (–0.422, –0.066, 1.016, 0.870, –0.142±2) (–0.422, –0.066, 1.016, 0.870±2, –0.142) (–0.422, –0.066, 1.016, 0.870±2, –0.142±2)	–1.389, –1.385, –0.676, 0.384±1.259i	指标-2鞍焦点
(0.422, 0.066, –1.016, –0.870, 0.142) (0.422, 0.066, –1.016, –0.870, 0.142±2) (0.422, 0.066, –1.016, –0.870±2, 0.142) (0.422, 0.066, –1.016, –0.870±2, 0.142±2)	–1.389, –1.385, –0.676, 0.384±1.259i	指标-2鞍焦点

下载: 导出CSV

表 3 双涡卷个数与p或q值之间的关系

p	吸引子含有多双涡卷个数	q	吸引子含有多双涡卷个数
0	1	0	2
1	3	1	4
2	5	2	6
···	···	···	···
p	2p+1	q	2q+2

下载: 导出CSV

表 4 模型2不同忆阻强度k₂对应的共存吸引子类别以及编号

忆阻强度k₂	共存对称吸引子类别	对应编号
–1.2	共存周期1吸引子	图8(a)
–0.8	共存周期2吸引子	图8(b)
–0.5	共存周期4吸引子	图8(c)
–0.1	共存单涡卷混沌吸引子	图8(d)

下载: 导出CSV

参考文献(30)

[1]	LIN Hairong, WANG Chunhua, CHEN Chengjie, 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: 10.1109/TCSI.2021.3081150.
[2]	WANG Chunhua, LI Yufei, YANG Gang, et al. A review of fractional-order chaotic systems of memristive neural networks[J]. Mathematics, 2025, 13(10): 1600. doi: 10.3390/math13101600.
[3]	BAO Bocheng, ZHOU Chunlong, BAO Han, et al. Heterogeneous Hopfield neural network with analog implementation[J]. Chaos, Solitons & Fractals, 2025, 194: 116234. doi: 10.1016/j.chaos.2025.116234.
[4]	DENG Quanli, WANG Chunhua, SUN Yichuang, et al. Delay difference feedback memristive map: Dynamics, hardware implementation, and application in path planning[J]. IEEE Transactions on Circuits and Systems I: Regular Papers, 2025, 72(12): 7993–8003. doi: 10.1109/TCSI.2025.3571961.
[5]	LUO Min, WANG Pingli, QIU Da, et al. Analysis and application of conditionally symmetric memristive chaotic systems with attractor growth phenomena[J]. Chaos, Solitons & Fractals, 2025, 200: 117027. doi: 10.1016/j.chaos.2025.117027.
[6]	赖强, 王君. 基于滑模趋近律的忆阻混沌系统有限和固定时间同步[J]. 物理学报, 2024, 73(18): 180503. doi: 10.7498/aps.73.20241013. LAI Qiang and WANG Jun. Finite and fixed-time synchronization of memristive chaotic systems based on sliding mode reaching law[J]. Acta Physica Sinica, 2024, 73(18): 180503. doi: 10.7498/aps.73.20241013.
[7]	赖强, 秦铭宏. 基于忆阻电磁辐射的极简环形HNN动力学行为增强研究[J]. 贵州师范大学学报: 自然科学版, 2025, 43(3): 1–11. doi: 10.16614/j.gznuj.zrb.2025.03.001. LAI Qiang and QIN Minghong. Enhanced dynamics of an extremely simple cyclic HNN based on memristive electromagnetic radiation[J]. Journal of Guizhou Normal University: Natural Sciences, 2025, 43(3): 1–11. doi: 10.16614/j.gznuj.zrb.2025.03.001.
[8]	贾美美, 曹佳伟, 白明明. 新型忆阻耦合异质神经元的放电模式和预定义时间混沌同步[J]. 物理学报, 2024, 73(17): 170502. doi: 10.7498/aps.73.20240872. JIA Meimei, CAO Jiawei, and BAI Mingming. Firing modes and predefined-time chaos synchronization of novel memristor-coupled heterogeneous neuron[J]. Acta Physica Sinica, 2024, 73(17): 170502. doi: 10.7498/aps.73.20240872.
[9]	ZHANG Sen, LI Yongxin, WANG Xiaoping, et al. Initial offset-boosted coexisting hidden chaos and firing multistability in memristive ring neural network with hardware implementation[J]. IEEE Transactions on Industrial Electronics, 2025, 72(2): 2024–2033. doi: 10.1109/TIE.2024.3429616.
[10]	DOUBLA I S, RAMAKRISHNAN B, TABEKOUENG Z N, et al. Infinitely many coexisting hidden attractors in a new hyperbolic-type memristor-based HNN[J]. The European Physical Journal Special Topics, 2022, 231(11): 2371–2385. doi: 10.1140/epjs/s11734-021-00372-x.
[11]	BIAMOU A L M, TAMBA V K, TAGNE F K, et al. Fractional-order-induced symmetric multi-scroll chaotic attractors and double bubble bifurcations in a memristive coupled Hopfield neural networks[J]. Chaos, Solitons & Fractals, 2024, 178: 114267. doi: 10.1016/j.chaos.2023.114267.
[12]	DENG Quanli, WANG Chunhua, SUN Yichuang, et al. Discrete memristive conservative chaotic map: Dynamics, hardware implementation, and application in secure communication[J]. IEEE Transactions on Cybernetics, 2025, 55(8): 3926–3934. doi: 10.1109/TCYB.2025.3565333.
[13]	武花干, 边逸轩, 陈墨, 等. 忆阻耦合异构忆阻细胞神经网络的多稳态与相位同步研究[J]. 电子与信息学报, 2024, 46(9): 3818–3826. doi: 10.11999/JEIT240010. WU Huagan, BIAN Yixuan, CHEN Mo, et al. Multistable state and phase synchronization of memristor-coupled heterogeneous memristive cellular neural network[J]. Journal of Electronics & Information Technology, 2024, 46(9): 3818–3826. doi: 10.11999/JEIT240010.
[14]	丁大为, 卢小齐, 胡永兵, 等. 分数阶忆阻耦合异质神经元的多稳态及硬件实现[J]. 物理学报, 2022, 71(23): 230501. doi: 10.7498/aps.71.20221525. DING Dawei, LU Xiaoqi, HU Yongbing, et al. Multistability of fractional-order memristor-coupled heterogeneous neurons and its hardware realization[J]. Acta Physica Sinica, 2022, 71(23): 230501. doi: 10.7498/aps.71.20221525.
[15]	赖强, 秦铭宏. 极简环形忆阻混沌神经网络的动力学分析与同步控制[J]. 电子与信息学报, 2025, 47(9): 3262–3273. doi: 10.11999/JEIT250212. LAI Qiang and QIN Minghong. Dynamic analysis and synchronization control of extremely simple cyclic memristive chaotic neural network[J]. Journal of Electronics & Information Technology, 2025, 47(9): 3262–3273. doi: 10.11999/JEIT250212.
[16]	BAO Han, HUA Mengjie, MA Jun, et al. Offset-control plane coexisting behaviors in two-memristor-based Hopfield neural network[J]. IEEE Transactions on Industrial Electronics, 2023, 70(10): 10526–10535. doi: 10.1109/TIE.2022.3222607.
[17]	WANG Chunhua, LIANG Junhui, and DENG Quanli. Dynamics of heterogeneous Hopfield neural network with adaptive activation function based on memristor[J]. Neural Networks, 2024, 178: 106408. doi: 10.1016/j.neunet.2024.106408.
[18]	LI Xuxin, LUO Min, ZHANG Bo, et al. Dynamic analysis and implementation of a multi-stable Hopfield neural network[J]. Chaos, Solitons & Fractals, 2025, 199: 116657. doi: 10.1016/j.chaos.2025.116657.
[19]	LI Haoyu, WANG Leimin, WAN Xiongbo, et al. Error transmission of chaos-based image encryption: Application to smart grid[J]. IEEE Transactions on Industrial Informatics, 2025, 21(12): 9889–9897. doi: 10.1109/TII.2025.3609139.
[20]	WU Chaojun, GUO Junxuan, YANG Ningning, et al. Analysis of piecewise-linear Hopfield neural network model based on a novel fractional-order memristor and its application to image encryption[J]. International Journal of Bifurcation and Chaos, 2025, 35(6): 2550065. doi: 10.1142/S0218127425500658.
[21]	L I Yongxin, LI Chunbiao, LEI Tengfei, et al. Offset boosting-entangled complex dynamics in the memristive Rulkov neuron[J]. IEEE Transactions on Industrial Electronics, 2024, 71(8): 9569–9579. doi: 10.1109/TIE.2023.3325558.
[22]	王雷敏, 程佳俊, 胡成, 等. 基于改进的忆阻器在字符联想记忆中的应用[J]. 电子与信息学报, 2023, 45(7): 2667–2674. doi: 10.11999/JEIT220709. WANG Leimin, CHENG Jiajun, HU Cheng, et al. Application of improved memristor in character associative memory[J]. Journal of Electronics & Information Technology, 2023, 45(7): 2667–2674. doi: 10.11999/JEIT220709.
[23]	WANG Chunhua, TANG Dong, LIN Hairong, et al. High-dimensional memristive neural network and its application in commercial data encryption communication[J]. Expert Systems with Applications, 2024, 242: 122513. doi: 10.1016/j.eswa.2023.122513.
[24]	BAO Han, DING Ruoyu, LIU Xiaofeng, et al. Memristor-cascaded Hopfield neural network with attractor scroll growth and STM32 hardware experiment[J]. Integration, 2024, 96: 102164. doi: 10.1016/j.vlsi.2024.102164.
[25]	LI Yongxin, LI Chunbiao, ZHANG Sen, et al. Offset boosting-oriented construction of multi-scroll attractor via a memristor model[J]. IEEE Transactions on Circuits and Systems I: Regular Papers, 2025, 72(2): 918–931. doi: 10.1109/TCSI.2024.3455350.
[26]	ZHANG Sen, CHEN Chengjie, ZHANG Yunzhen, et al. Multidirectional multidouble-scroll hopfield neural network with application to image encryption[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2025, 55(1): 735–746. doi: 10.1109/TSMC.2024.3489226.
[27]	LAI Qiang, WAN Zhiqiang, ZHANG Hui, 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, 2023, 34(10): 7824–7837. doi: 10.1109/TNNLS.2022.3146570.
[28]	LI Jianghao, WANG Chunhua, and DENG Quanli. Symmetric multi-double-scroll attractors in Hopfield neural network under pulse controlled memristor[J]. Nonlinear Dynamics, 2024, 112(16): 14463–14477. doi: 10.1007/s11071-024-09791-6.
[29]	WAN Qiuzhen, CHEN Simiao, YANG Qiao, et al. Grid multi-scroll attractors in memristive Hopfield neural network under pulse current stimulation and multi-piecewise memristor[J]. Nonlinear Dynamics, 2023, 111(19): 18505–18521. doi: 10.1007/s11071–023-08834–8.
[30]	TANG Dong, WANG Chunhua, LIN Hairong, et al. Dynamics analysis and hardware implementation of multi-scroll hyperchaotic hidden attractors based on locally active memristive Hopfield neural network[J]. Nonlinear Dynamics, 2024, 112(2): 1511–1527. doi: 10.1007/s11071-023-09128-9.