Eur. Phys. J. Spec. Top. (2022) 231: 1801-1814
https://doi.org/10.1140/epjs/s11734-022-00456-2

Regular Article

Anti-synchronization of a M-Hopfield neural network with generalized hyperbolic tangent activation function

¹ Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca, MOR, México
² CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca, MOR, México
³ Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp-IICBA), Av. Universidad No. 1001, Col Chamilpa, C.P. 62209, Cuernavaca, MOR, México

^b jose.ga@cenidet.tecnm.mx

Received: 26 March 2021
Accepted: 13 January 2022
Published online: 29 January 2022

Abstract

This paper analyzes non-integer Hopfield neural network dynamics introducing the hyperbolic tangent transfer function generalized by the Mittag-Leffler function and the M-truncated derivative with constant and variable order. The novel neural network’s (ANN) behaviors are studied through their dynamics depicted in phase portraits and the 0-1 test to determine where the ANN displays strong chaotic behaviors. According to the numerical results, the generalized Hopfield (M-HNTF) reveals weak chaotic dynamics with constant values under 0.99 and regular behaviors lower than 0.8. Considering the variable order, the chaotic behaviors depend on the decay rate of the time-varying function. Due to this, we got systems with weak chaotic dynamics until strong chaotic dynamics. Next, we used two scenarios to anti-synchronize a system master and a slave system. The first considering a dynamic, chaotic system and a regular system, the second: two M-HNTF with variable order. Numerical results illustrate those mentioned above, showing the control aim. Getting new chaotic dynamics from non-integer systems with variable order is essential to develop protocols to offer secure communications, new random number generators, image encrypts schemes, to name a few.