Mean square stability and almost sure exponential stability of two step Maruyama methods of stochastic delay Hopfield neural networks

Abstract

In this paper, the two-step Maruyama methods of stochastic delay Hopfield neural networks are studied. We have found that under what choices of step-size, the two-step Maruyama methods of stochastic delay Hopfield networks, maintain the stability of exact solutions. The mean-square stability of two-step Maruyama methods of stochastic delay Hopfield neural networks is investigated under suitable conditions. Also, the almost sure exponential stability of two-step Maruyama methods of stochastic delay Hopfield networks is proved using the semi-martingale convergence theorem. Further, the comparisons of stability conditions to the previous results in Liu and Zhu (2015), Rathinasamy (2012) and Ronghua et al. (2010) are given. Numerical experiments are provided to illustrate our theoretical results.