University of Tabriz Computational Methods for Differential Equations 2345-3982 12 1 2024 11 01 Almost sure exponential numerical stability of balanced Maruyama with two step approximations of stochastic time delay Hopfield neural networks 136 148 16455 10.22034/cmde.2023.55861.2330 EN Sivarajan Kopperundevi Department of Mathematics, Dr. M.G.R Educational and Research Institute(To be Deemed), Maduravoyal, Chennai-600 095, India. Journal Article 2023 03 15 This study examines the balanced Maruyama with two step approximations of stochastic Hopfield neural networks with delay. The main aim of this paper is to discover the conditions under which the exact solutions remain stable for the balanced Maruyama with two-step approximations of stochastic delay Hopfield neural networks (SDHNN). The semi martingale theorem for convergence is used to demonstrate the almost sure exponential stability of balanced Maruyama with two-step approximations of stochastic delay Hopfield networks. Additionally, the numerical balanced Euler approximation's stability conditions are compared. Our theoretical findings are illustrated with numerical experiments. Almost sure exponential stability balanced two step Maruyama numerical approximations Hopfield neural networks Stochastic delay differential equations https://cmde.tabrizu.ac.ir/article_16455_1d9d27cd91712d1c8b35c7161b3fea7a.pdf