Chaotic dynamics in Sprott's memristive artificial neural network: dynamic analysis, circuit implementation and synchronization

M. I. Kopp; I. Samuilik; M. I. Kopp; I. Samuilik

doi:10.3934/math.2025860

AIMS Mathematics

2025, Volume 10, Issue 8: 19240-19266. doi: 10.3934/math.2025860

Previous Article Next Article

Research article Special Issues

Chaotic dynamics in Sprott's memristive artificial neural network: dynamic analysis, circuit implementation and synchronization

M. I. Kopp ¹,
I. Samuilik ^{2,3
,
,}

1.
Institute for Single Crystals, NAS of Ukraine, Nauky Ave. 60, Kharkiv 61072, Ukraine
2.
Institute of Life Sciences and Technologies, Daugavpils University, 13 Vienibas Street, LV-5401 Daugavpils, Latvia
3.
Institute of Applied Mathematics, Riga Technical University, LV-1048 Riga, Latvia

Received: 31 March 2025 Revised: 15 July 2025 Accepted: 21 July 2025 Published: 25 August 2025
MSC : 34A34, 37D45, 37G35, 37H15, 68T01

The paper presents a new artificial neural network (ANN) obtained by embedding a memristor into the self-connection synapse of one neuron from the Sprott ANN. The mathematical model of the new network is described by a five-dimensional (5D) dynamic system. A comprehensive analysis of its dynamic properties is carried out, including bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, timing diagrams and phase portraits, multistability, and offset boosting control. The theoretical model is further verified by electronic simulation of a chaotic system using Multisim software. A synchronization model of two coupled memristive subneural Sprott networks is proposed to simulate synchronization between regions of biological neural systems. Linearization methods and Lyapunov stability theory are employed to prove synchronization. These results provide useful insights into the nonlinear dynamic characteristics of the new Sprott ANN.
- artificial neural networks (ANNs),
- memristor,
- chaotic behavior,
- circuit implementation,
- synchronization
Citation: M. I. Kopp, I. Samuilik. Chaotic dynamics in Sprott's memristive artificial neural network: dynamic analysis, circuit implementation and synchronization[J]. AIMS Mathematics, 2025, 10(8): 19240-19266. doi: 10.3934/math.2025860

Related Papers:

Abstract

The paper presents a new artificial neural network (ANN) obtained by embedding a memristor into the self-connection synapse of one neuron from the Sprott ANN. The mathematical model of the new network is described by a five-dimensional (5D) dynamic system. A comprehensive analysis of its dynamic properties is carried out, including bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, timing diagrams and phase portraits, multistability, and offset boosting control. The theoretical model is further verified by electronic simulation of a chaotic system using Multisim software. A synchronization model of two coupled memristive subneural Sprott networks is proposed to simulate synchronization between regions of biological neural systems. Linearization methods and Lyapunov stability theory are employed to prove synchronization. These results provide useful insights into the nonlinear dynamic characteristics of the new Sprott ANN.

References

[1]	A. A. Shukur, V.-T. Pham, V. K. Tamba, G. Grassi, Hyperchaotic oscillator with line and spherical equilibria: stability, entropy, and implementation for random number generation, Symmetry, 16 (2024), 1341. https://doi.org/10.3390/sym16101341 doi: 10.3390/sym16101341
[2]	Z. Li, S. Zhao, Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation, AIMS Mathematics, 9 (2024), 22590–22601. https://doi.org/10.3934/math.20241100 doi: 10.3934/math.20241100
[3]	K. Zhang, J. P. Cao, Dynamic behavior and modulation instability of the generalized coupled fractional nonlinear Helmholtz equation with cubic-quintic term, Open Phys., 23 (2025), 20250144. https://doi.org/10.1515/phys-2025-0144 doi: 10.1515/phys-2025-0144
[4]	T. Bohr, M. H. Jensen, G. Paladin, A. Vulpiani, Dynamical systems approach to turbulence, 2 Eds., Cambridge: Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511599972
[5]	S. Soldatenko, A. Bogomolov, A. Ronzhin, Mathematical modelling of climate change and variability in the context of outdoor ergonomics, Mathematics, 9 (2021), 2920. https://doi.org/10.3390/math9222920 doi: 10.3390/math9222920
[6]	O. Kozlovska, F. Sadyrbaev, I. Samuilik, A new 3D chaotic attractor in gene regulatory network, Mathematics, 12 (2024), 100. https://doi.org/10.3390/math12010100 doi: 10.3390/math12010100
[7]	J. Demongeot, F. Sadyrbaev, I. Samuilik, Mathematical modeling of gene networks, Front. Appl. Math. Stat., 10 (2024), 1514380. https://doi.org/10.3389/fams.2024.1514380 doi: 10.3389/fams.2024.1514380
[8]	X. X. Yin, J. Chen, W. X. Yu, Y. Huang, W. X. Wei, X. J. Xiang, et al., Five-dimensional memristive Hopfield neural network dynamics analysis and its application in secure communication, Circuit World, 50 (2024), 67–81. https://doi.org/10.1108/cw-05-2022-0135 doi: 10.1108/cw-05-2022-0135
[9]	A. Ipatovs, I. C. Victor, D. Pikulins, S. Tjukovs, A. Litvinenko, Complete bifurcation analysis of the vilnius chaotic oscillator, Electronics, 12 (2023), 2861. https://doi.org/10.3390/electronics12132861 doi: 10.3390/electronics12132861
[10]	S. Tjukovs, D. Surmacs, J. Grizans, C. V. Iheanacho, D. Pikulins, Implementation of buck DC-DC converter as built-in chaos generator for secure IoT, Electronics, 13 (2024), 20. https://doi.org/10.3390/electronics13010020 doi: 10.3390/electronics13010020
[11]	S. Hashemi, M. A. Pourmina, S. Mobayen, M. R. Alagheband, Multiuser wireless speech encryption using synchronized chaotic systems, Int. J. Speech. Technol., 24 (2021), 651–663. https://doi.org/10.1007/s10772-021-09821-3 doi: 10.1007/s10772-021-09821-3
[12]	K. U. Shahna, Novel chaos based cryptosystem using four-dimensional hyper chaotic map with efficient permutation and substitution techniques, Chaos Soliton. Fract., 170 (2023), 113383. https://doi.org/10.1016/j.chaos.2023.113383 doi: 10.1016/j.chaos.2023.113383
[13]	I. N. da Silva, D. H. Spatti, R. A. Flauzino, L. H. Bartocci Liboni, S. F. dos Reis Alves, Artificial neural networks: A practical course, Cham: Springer, 2017. https://doi.org/10.1007/978-3-319-43162-8
[14]	M. Minsky, S. A. Papert, Perceptrons: An introduction to computational geometry, Cambridge: The MIT Press, 2017. https://doi.org/10.7551/mitpress/11301.001.0001
[15]	A. Baliyan, K. Gaurav, S. K. Mishra, A review of short term load forecasting using artificial neural network models, Procedia Computer Science, 48 (2015), 121–125. https://doi.org/10.1016/j.procs.2015.04.160 doi: 10.1016/j.procs.2015.04.160
[16]	J. L. Hindmarsh, R. M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 296 (1982), 162–164. https://doi.org/10.1038/296162a0 doi: 10.1038/296162a0
[17]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445–466. https://doi.org/10.1016/S0006-3495(61)86902-6 doi: 10.1016/S0006-3495(61)86902-6
[18]	J. Nagumo, S. Arimoto, S. Yoshizawa, An Active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061–2070. https://doi.org/10.1109/JRPROC.1962.288235 doi: 10.1109/JRPROC.1962.288235
[19]	A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500–544. https://doi.org/10.1113/jphysiol.1952.sp004764 doi: 10.1113/jphysiol.1952.sp004764
[20]	Z. T. Njitacke, S. Parthasarathy, C. N. Takembo, K. Rajagopal, J. Awrejcewicz, Dynamics of a memristive FitzHugh-Rinzel neuron model: application to information patterns, Eur. Phys. J. Plus, 138 (2023), 473. https://doi.org/10.1140/epjp/s13360-023-04120-z doi: 10.1140/epjp/s13360-023-04120-z
[21]	C. X. Wang, H. J. Cao, Parameter space of the Rulkov chaotic neuron model, Commun. Nonlinear Sci., 19 (2014), 2060–2070. https://doi.org/10.1016/j.cnsns.2013.10.004 doi: 10.1016/j.cnsns.2013.10.004
[22]	J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. U.S.A., 81 (1984), 3088–3092. https://doi.org/10.1073/pnas.81.10.3088 doi: 10.1073/pnas.81.10.3088
[23]	H. R. Lin, C. H. Wang, Q. L. Deng, C. Xu, Z. K. Deng, C. Zhou, Review on chaotic dynamics of memristive neuron and neural network, Nonlinear Dyn., 106 (2021), 959–973. https://doi.org/10.1007/s11071-021-06853-x doi: 10.1007/s11071-021-06853-x
[24]	H. R. Lin, C. H. Wang, F. Yu, J. R. Sun, S. C. Du, Z. K. Deng, et al., A review of chaotic systems based On memristive hopfield neural networks, Mathematics, 11 (2023), 1369. https://doi.org/10.3390/math11061369 doi: 10.3390/math11061369
[25]	L. Chua, Memristor-The missing circuit element, IEEE Transactions on Circuit Theory, 18 (1971), 507–519. http://doi.org/10.1109/TCT.1971.1083337 doi: 10.1109/TCT.1971.1083337
[26]	D. B. Strukov, G. S. Snider, D. R. Stewart, R. S. Williams, The missing memristor found, Nature, 453 (2008), 80–83. http://doi.org/10.1038/nature06932 doi: 10.1038/nature06932
[27]	Y. C. Huang, J. X. Liu, J. Harkin, L. McDaid, Y. L. Luo, An memristor-based synapse implementation using BCM learning rule, Neurocomputing, 423 (2021), 336–342. https://doi.org/10.1016/j.neucom.2020.10.106 doi: 10.1016/j.neucom.2020.10.106
[28]	B. W. Yan, G. J. Wang, J. G. Yu, X. Z. Jin, H. L. Zhang, Spatial-temporal Chebyshev graph neural network for traffic flow prediction in IoT-based ITS, IEEE Internet Things, 9 (2022), 9266–9279. https://doi.org/10.1109/JIOT.2021.3105446 doi: 10.1109/JIOT.2021.3105446
[29]	L. Alzubaidi, J. L. Zhang, A. J. Humaidi, A. Al-Dujaili, Y. Duan, O. Al-Shamma, et al., Review of deep learning: Concepts, CNN architectures, challenges, applications, future directions, J. Big Data, 8 (2021), 53. https://doi.org/10.1186/s40537-021-00444-8 doi: 10.1186/s40537-021-00444-8
[30]	K. Yamazaki, V. K. Vo-Ho, B. Darshan, N. Le, Spiking neural networks and their applications: A review, Brain Sci., 12 (2022), 863. https://doi.org/10.3390/brainsci12070863 doi: 10.3390/brainsci12070863
[31]	O. I. Abiodun, A. Jantan, A. E. Omolara, K. V. Dada, A. M. Umar, et al., Comprehensive review of artificial neural network applications to pattern recognition, IEEE Access, 7 (2019), 158820–158846. https://doi.org/10.1109/ACCESS.2019.2945545 doi: 10.1109/ACCESS.2019.2945545
[32]	J. Vohradsky, Neural network model of gene expression, FASEB J., 15 (2001), 846–854. https://doi.org/10.1096/fj.00-0361com doi: 10.1096/fj.00-0361com
[33]	S. Haykin, Neural networks: A comprehensive foundation, New Jersey: Prentice-Hall, 1998.
[34]	A. Das, A. B. Roy, P. Das, Chaos in a three dimensional neural network, Appl. Math. Model., 24 (2000), 511–522. https://doi.org/10.1016/S0307-904X(99)00046-3 doi: 10.1016/S0307-904X(99)00046-3
[35]	O. Kozlovska, F. Sadyrbaev, On attractors in systems of ordinary differential equations arising in models of genetic networks, Vibroengineering PROCEDIA, 49 (2023), 136–140. https://doi.org/10.21595/vp.2023.23343 doi: 10.21595/vp.2023.23343
[36]	O. Kozlovska, F. Sadyrbaev, I. Samuilik, A new 3D chaotic attractor in gene regulatory network, Mathematics, 12 (2024), 100. https://doi.org/10.3390/math12010100 doi: 10.3390/math12010100
[37]	J. C. Sprott, Chaotic dynamics on large networks, Chaos, 18 (2008), 023135. https://doi.org/10.1063/1.2945229 doi: 10.1063/1.2945229
[38]	K. Funahashi, Y. Nakamura, Approximation of dynamical systems by continuous time recurrent neural networks, Neural Networks, 6 (1993), 801–806. https://doi.org/10.1016/s0893-6080(05)80125-x doi: 10.1016/s0893-6080(05)80125-x
[39]	J. C. Sprott, Elegant chaos algebraically simple chaotic flows, Singapore: World Scientific Publishing Company, 2010. https://doi.org/10.1142/7183
[40]	I. Samuilik, F. Sadyrbaev, D. Ogorelova, Comparative analysis of models of gene and neural networks, Contemp. Math., 4 (2023), 217–229. https://doi.org/10.37256/cm.4220232404 doi: 10.37256/cm.4220232404
[41]	D. Ogorelova, F. Sadyrbaev, Remarks on the mathematical modeling of gene and neuronal networks by ordinary differential equations, Axioms, 13 (2024), 61. https://doi.org/10.3390/axioms13010061 doi: 10.3390/axioms13010061
[42]	I. Samuilik, F. Sadyrbaev, S. Atslega, On mathematical models of artificial neural networks, The 22nd International Scientific Conference "Engineering for Rural Development", Latvija, Jelgava, 2023, 45–50.
[43]	I. Samuilik, A mathematical model for a class of networks in applications, Phd Thesis, Daugavpils University, 2023.
[44]	Q. L. Deng, C. H. Wang, H. R. Lin, Memristive Hopfield neural network dynamics with heterogeneous activation functions and its application, Chaos Soliton. Fract., 178 (2024), 114387. https://doi.org/10.1016/j.chaos.2023.114387 doi: 10.1016/j.chaos.2023.114387
[45]	G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems: A method for computing all of them, Meccanica, 15 (1980), 9–20. https://doi.org/10.1007/BF02128236 doi: 10.1007/BF02128236
[46]	A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9 doi: 10.1016/0167-2789(85)90011-9
[47]	H. Binous, N. Zakia, An improved method for Lyapunov exponents computation, Wolfram Library Archive, 2008. Available from: https://library.wolfram.com/infocenter/ID/7109/.
[48]	M. Sandri, Numerical calculation of Lyapunov exponents, The Mathematica Journal, 6 (1996), 78–84.
[49]	C. P. Silva, Shil'nikov's theorem: A tutorial, IEEE T. Circuits, 40 (1993), 675–682. https://doi.org/10.1109/81.246142 doi: 10.1109/81.246142
[50]	P. Frederickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, The Liapunov dimension of strange attractors, J. Differ. Equations, 49 (1983), 185–207. https://doi.org/10.1016/0022-0396(83)90011-6 doi: 10.1016/0022-0396(83)90011-6
[51]	J. J. Wen, Y. R. Feng, X. H. Tao, Y. H. Cao, Dynamical analysis of a new chaotic system: Hidden attractor, coexisting-attractors, offset boosting, and DSP realization, IEEE Access, 9 (2021), 167920–167927. https://doi.org/10.1109/ACCESS.2021.3136249 doi: 10.1109/ACCESS.2021.3136249
[52]	E. W. Weisstein, Routh-Hurwitz theorem, MathWorld: A wolfram web resource. Available from: https://mathworld.wolfram.com/Routh-HurwitzTheorem.html.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)