On stability of nonconformable Hopfield neural networks
Anes Moulai-Khatir
Received November 24, 2024. Published online April 8, 2025.
Abstract: This work focuses on a particular type of Hopfield neural network that generalizes classical fractional derivatives and is distinguished by nonconformable fractional derivatives. The main goal is to determine the basic characteristics of these networks, such as the circumstances under which equilibrium points are present and distinct. By demonstrating the exponential stability of the network, we further investigate its behavior and rigorously deduce these criteria. This is accomplished by constructing a Lyapunov function, a potent instrument frequently used in stability studies. In addition to verifying the obtained stability constraints, the theoretical conclusions are supported by comprehensive numerical simulations that show the dynamics of the neural network in a variety of circumstances. These simulations provide specific illustrations of how the network reacts to various parameter combinations and inputs. Overall, this work contributes to the understanding of neural networks with fractional-order dynamics, offering insights into their mathematical properties and potential applications in areas requiring robust and stable systems.
References: [1] T. Abdeljawad: On conformable fractional calculus. J. Comput. Appl. Math. 279 (2015), 57-66. DOI 10.1016/j.cam.2014.10.016 | MR 3293309 | Zbl 1304.26004
[2] M. U. Akhmet, M. Karacaören: A Hopfield neural network with multi-compartmental activation. Neural Comput. Appl. 29 (2018), 815-822. DOI 10.1007/s00521-016-2597-9
[3] K.-S. Chiu: Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument. Abstr. Appl. Anal. 2013 (2013), Article ID 196139, 13 pages. DOI 10.1155/2013/196139 | MR 3147863 | Zbl 1298.34136
[4] K.-S. Chiu: Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument. Acta Appl. Math. 151 (2017), 199-226. DOI 10.1007/s10440-017-0108-3 | MR 3694762 | Zbl 1373.92010
[5] N. Echi, F. Mabrouk, F. Omri: Exponential stability of non-conformable fractional-order systems. J. Appl. Anal. 30 (2024), 407-415. DOI 10.1515/jaa-2023-0134 | MR 4831787 | Zbl 07959851
[6] K. Gopalsamy: Stability of artificial neural networks with impulses. Appl. Math. Comput. 154 (2004), 783-813. DOI 10.1016/S0096-3003(03)00750-1 | MR 2072820 | Zbl 1058.34008
[7] P. M. Guzmán, G. Langton, L. M. Lugo Motta Bittencurt, J. Medina, J. E. Nápoles Valdes: A new definition of a fractional derivative of local type. J. Math. Anal. 9 (2018), 88-98. MR 3797479
[8] P. M. Guzmán, J. E. Nápoles Valdés: A note on the oscillatory character of some non conformable generalized Liénard system. Adv. Math. Models Appl. 2 (2019), 127-133.
[9] J. J. Hopfield: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79 (1982), 2554-2558. DOI 10.1073/pnas.79.8.2554 | MR 652033 | Zbl 1369.92007
[10] J. J. Hopfield: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81 (1984), 3088-3092. DOI 10.1073/pnas.81.10.3088 | Zbl 1371.92015
[11] S. Kasmi, F. Mabrouk, F. Omri: Exponential stabilization of conformable fractional bilinear systems with multiple inputs. Asian-Eur. J. Math. 16 (2023), Article ID 2350169, 10 pages. DOI 10.1142/S1793557123501693 | MR 4641738 | Zbl 07955874
[12] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh: A new definition of fractional derivative. J. Comput. Appl. Math. 264 (2014), 65-70. DOI 10.1016/j.cam.2014.01.002 | MR 3164103 | Zbl 07955874
[13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). DOI 10.1016/s0304-0208(06)x8001-5 | MR 2218073 | Zbl 1092.45003
[14] A. Kütahyalioglu, F. Karakoç: Exponential stability of Hopfield neural networks with conformable fractional derivative. Neurocomputing 456 (2021), 263-267. DOI 10.1016/j.neucom.2021.05.076
[15] F. Mabrouk: Homogeneity-based exponential stability analysis for conformable fractional-order systems. Ukr. Math. J. 75 (2024), 1590-1600. DOI 10.1007/s11253-024-02280-4 | MR 4747111 | Zbl 1541.34017
[16] F. Mabrouk: Stabilization of homogeneous conformable fractional-order systems. Ukr. Mat. Zh. 76 (2024), 1802-1812. DOI 10.3842/umzh.v76i12.7689
[17] W. McCulloch, W. Pitts: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5 (1943), 115-133. DOI 10.1007/BF02478259 | MR 0010388 | Zbl 0063.03860
[18] A. Moulai-Khatir, A. Cherraf: On asymptotics of some conformable differential equations. J. Fract. Calc. Appl. 14 (2023), 147-156. DOI 10.21608/jfca.2023.284192 | MR 4632492 | Zbl 07890988
[19] I. Podlubny: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198. Academic Press, San Diego (1999). MR 1658022 | Zbl 0924.34008
[20] F. Rosenblatt: The perceptron: A probabilistic model for information storage and organization in the brain. Psychol. Rev. 65 (1958), 386-408. DOI 10.1037/h0042519 | MR 4679172 | Zbl 1529.68035
[21] F. Rosenblatt: Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Spartan, Washington (1962). MR 0135635 | Zbl 0143.43504
[22] M. Vivas-Cortez, J. Nápoles-Valdés, J. E. Hernández Hernández, J. Velasco Velasco, O. Larreal: On non conformable fractional Laplace transform. Appl. Math. Inf. Sci. 15 (2021), 403-409. DOI 10.18576/amis/150401 | MR 4306237
Affiliations: Anes Moulai-Khatir, Department of Industrial Engineering, Institute of Maintenance and Industrial Safety, University of Oran 2, B.P 1015 El M'naouer 31000 Oran, Algeria, and Laboratory of Nonlinear Analysis and Applied Mathematics, University of Tlemcen, 22, Rue Abi Ayed Abdelkrim Fg Pasteur, B.P 119 13000, Tlemcen, Algeria, e-mail: anes.mkh@gmail.com, moulaikhatir.anes@univ-oran2.dz
