Matignon-Based Stability and Weight Synchronization of a Fractional Time Delay Neural Network Model
Department of Mathematics, City University of Science and Information Technology (CUSIT), Peshawar 25000, Pakistan
| Abstract: | Artificial neural networks (ANNs) are powerful models inspired by the structure and function of the human brain. They are widely used for tasks such as classification, prediction, and model recognition. This study examines the stability of fractional-order neural networks with neuronal conditions, dynamic behavior, synchronization, and delays of time "σ" . Synchronization and stability for delayed neural network models are two important aspects of dynamic behavior. For a calculated fractional-order, the state of the state variable wi(t) are synchronized with each other. Weight synchronization of wi (i=1, 2, 3,..., 6) provides coherent updates during training, helping neural networks to study stable models. The incommensurate fractional-orders are linked to a system where each dynamic component develops with a different value, i.e. qi ≠ qj (i≠j) is inconsistent. These fractional-orders are calculated for the system’s eigenvalues and their singular points within the stability region defined by the Matignon-based stability. As the time delay decreases, more activation functions are induced, and the variable state of w4(t) requires longer relaxation times to be more stable than the variable state of w3(t). The Grunwald-Letnikov method is used to solve a fractional neural network system numerically and effectively handle fractional derivatives. This approach helps to more accurately simulate memory in neural networks. |
| Keywords: | Artificial Neural Networks; Stability Analysis; Synchronization; Time Delay; Dynamic Systems |
| DOI: | 10.57237/j.wjms.2026.01.002 |
View:
Downloads:
Received: 8 October 2025
Accepted: 28 November 2025
Published Online: 14 January 2026
References
| [1] | Li, Y., & Shen, S. (2020). Almost automorphic solutions for Clifford-valued neutral-type fuzzy cellular neural networks with leakage delays on time scales. Neurocomputing, 417, 23-35. |
| [2] | Xiu, C., Zhou, R., & Liu, Y. (2020). New chaotic memristive cellular neural network and its application in a secure communication system. Chaos, Solitons & Fractals, 141, 110316. |
| [3] | Ji, L., Chang, M., Shen, Y., & Zhang, Q. (2020). Recurrent convolutions of binary-constrained cellular neural network for texture recognition. Neurocomputing, 387, 161- 171. |
| [4] | Kumar, R., & Das, S. (2020). Exponential stability of inertial BAM neural network with time-varying impulses and mixed time-varying delays via matrix measure approach. Communications in Nonlinear Science and Numerical Simulation, 81, 105016. |
| [5] | Xu, C., Liao, M., Li, P., Liu, Z., & Yuan, S. (2021). New results on pseudo almost periodic solutions of quaternion-valued fuzzy cellular neural networks with delays. Fuzzy Sets and Systems, 411, 25-47. |
| [6] | Kobayashi, M. (2021). Complex-valued Hopfield neural networks with real weights in synchronous mode. Neurocomputing, 423, 535-540. |
| [7] | Cui, W., Wang, Z., & Jin, W. (2021). Fixed-time synchronization of Markovian jump fuzzy cellular neural networks with stochastic disturbance and time-varying delays. Fuzzy Sets and Systems, 411, 68-84. |
| [8] | Huang, C., Su, R., Cao, J., & Xiao, S. (2020). Asymptotically stable high-order neutral cellular neural networks with proportional delays and D-operators. Mathematics and Computers in Simulation, 171, 127-135. |
| [9] | Meng, B., Wang, X., Zhang, Z., & Wang, Z. (2020). Necessary and sufficient conditions for normalization and sliding mode control of singular fractional-order systems with uncertainties. Science China Information Sciences, 63, 1-10. 20. |
| [10] | Hsu, C. H., & Lin, J. J. (2019). Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays. Journal of Mathematical Analysis and Applications, 470(1), 388-400. |
| [11] | Tang, R., Yang, X., & Wan, X. (2019). Finite-time cluster synchronization for a class of fuzzy cellular neural networks via non-chattering quantized controllers. Neural Networks, 113, 79-90. |
| [12] | Wang, W. (2018). Finite-time synchronization for a class of fuzzy cellular neural networks with time-varying coefficients and proportional delays. Fuzzy Sets and Systems, 338, 40-49. |
| [13] | Wang, S., Zhang, Z., Lin, C., & Chen, J. (2021). Fixed-time synchronization for complex-valued BAM neural networks with time-varying delays via pinning control and adaptive pinning control. Chaos, Solitons & Fractals, 153, 111583. |
| [14] | Zhao, R., Wang, B., & Jian, J. (2022). Global stabilization of quaternion-valued inertial BAM neural networks with time-varying delays via time-delayed impulsive control. Mathematics and Computers in Simulation, 202, 223-245. |
| [15] | Li, Y., & Qin, J. (2018). Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays. Neurocomputing, 292, 91-103. |
| [16] | Kong, F., Zhu, Q., Wang, K., & Nieto, J. J. (2019). Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D-operator. Journal of the Franklin Institute, 356(18), 11605-11637. |
| [17] | Xu, C., & Zhang, Q. (2014). On the antiperiodic solutions for Cohen-Grossberg shunting inhibitory neural networks with time-varying delays and impulses. Neural Computation, 26(10), 2328-2349. |
| [18] | Ali, M. S., Yogambigai, J., Saravanan, S., & Elakkia, S. (2019). Stochastic stability of neutral-type Markovian-jumping BAM neural networks with time-varying delays. Journal of Computational and Applied Mathematics, 349, 142-156. |
| [19] | Cong, E. Y., Han, X., & Zhang, X. (2020). Global exponential stability analysis of discrete-time BAM neural networks with delays: A mathematical induction approach. Neurocomputing, 379, 227-235. |
| [20] | Ayachi, M. (2022). Measure-pseudo almost periodic dynamical behaviors for BAM neural networks with D operator and hybrid time-varying delays. Neurocomputing, 486, 160-173. |
| [21] | Shi, J., He, K., & Fang, H. (2022). Chaos, Hopf bifurcation, and control of a fractional-order delay financial system. Mathematics and Computers in Simulation, 194, 348-364. |
| [22] | Xiao, J., Wen, S., Yang, X., & Zhong, S. (2020). New approach to global Mittag-Leffler synchronization problem of fractional-order quaternion-valued BAM neural networks based on a new inequality. Neural Networks, 122, 320-337. |
| [23] | Xu, C., Mu, D., Pan, Y., Aouiti, C., Pang, Y., & Yao, L. (2022). Probing into bifurcation for fractional-order BAM neural networks concerning multiple time delays. Journal of Computational Science, 62, 101701. |
| [24] | Xu, C., Liao, M., Li, P., Guo, Y., & Liu, Z. (2021). Bifurcation properties for fractional-order delayed BAM neural networks. Cognitive Computation, 13, 322-356. |
| [25] | Wang, F., Yang, Y., Xu, X., & Li, L. (2017). Global asymptotic stability of impulsive fractional-order BAM neural networks with time delay. Neural Computing and Applications, 28, 345-352. |
| [26] | Ye, R., Liu, X., Zhang, H., & Cao, J. (2019). Global Mittag-Leffler synchronization for fractional-order BAM neural networks with impulses and multiple variable delays via delayed-feedback control strategy. Neural Processing Letters, 49, 1-18. |
| [27] | Xu, C., Liu, Z., Aouiti, C., Li, P., Yao, L., & Yan (2022). New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage 22 delays. Cognitive Neurodynamics, 16(5), 1233-1248. |
| [28] | Xiao, J., Guo, X., Li, Y., Wen, S., Shi, K., & Tang, Y (2022). Extended analysis on the global Mittag-Leffler synchronization problem for fractional-order octonion-valued BAM neural networks. Neural Networks, 154, 491-507. |
| [29] | Popa, C. A. (2023). Mittag-Laffler stability and synchronization of neutral-type fractional-order neural networks with leakage delay and mixed delays. Journal of the Franklin Institute, 360(1), 327-355. |
| [30] | Ci, J., Guo, Z., Long, H., Wen, S., & Huang, T. (2023). Multiple asymptotic periodicities of fractional-order delayed neural networks under state-dependent switching. Neural Networks, 157, 11-25. |
| [31] | Shah, D. K., Vyawahare, V. A., & Sadanand, S. (2025). Artificial neural network approximation of special functions: design, analysis, and implementation. International Journal of Dynamics and Control, 13(1), 1-23. |
| [32] | Admon, M. R., Senu, N., Ahmadian, A., Majid, Z. A., & Salahshour, S. (2024). A new and modern scheme for solving fractal-fractional differential equations based on a deep feedforward neural network with multiple hidden layers. Mathematics and Computers in Simulation, 218, 311-333. |
| [33] | Maurya, S. S., Kannan, J. B., Patel, K., Dutta, P., Biswas, K., Santhanam, M. S., & U. D. (2024). Asymmetric dynamical localization and precision measurement of the micromotion of a Bose-Einstein condensate. Physical Review A, 110(5), 053307. |
| [34] | Joshi, D. D., Bhalekar, S., & Gade, P. M. (2024). Stability analysis of fractional differential equations with delay. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(5). |
| [35] | Chettouh, B. (2024). Stability, Bifurcations and Control in Fractional-order Chaotic Systems (Doctoral dissertation, Université Mohamed Khider (Biskra-Algérie)). |
| [36] | Ur Rahman, H., Shuaib, M., Ismail, E. A., & Li, S. (2023). Enhancing medical ultrasound imaging through fractional mathematical modeling of ultrasound bubble dynamics. Ultrasonics Sonochemistry, 100(9), 106603. |
| [37] | Gopalsamy, K., & He, X. Z. (1994). Delay-dependent stability in bidirectional associative memory networks. IEEE Transactions on Neural Networks, 5(6), 998-1002. |
| [38] | Zhang, C., Zheng, B., & Wang, L. (2009). Multiple Hopf bifurcations of a symmetric BAM neural network model with delay. Applied Mathematics Letters, 22(4), 616-622. |
| [39] | Vinagre, B. M., Chen, Y. Q., & Petráš, I. (2003). Two direct Tustin discretization methods for a fractional-order differentiator/integrator. Journal of the Franklin Institute, 340(5), 349-362. |
Join the Editorial Board
We invite active, qualified and high profile scientists and researchers to join as Editorial Board Members.
Join UsBecome a Reviewer
Scholars with a strong interest in reviewing are invited to join the reviewer panel to ensure the quality of the research to be published.
Join Us