Strong convergence and almost sure exponential stability of balanced numerical approximations to stochastic delay Hopfield neural networks

Abstract

This paper deals with the balanced numerical schemes of the stochastic delay Hopfield neural networks. The balanced methods have a strong convergence rate of at least 12 and the balanced schemes have almost sure exponential stability under certain conditions. Under the Lipchitz and linear growth conditions, the balanced Euler methods are proved to have a strong convergence of order 12 in mean-square sense. Using the Lipchitz conditions on the various parameters of the model, based on the semimartingale convergence theorem and some reasonable assumptions, the balanced Euler methods of the stochastic delay Hopfield neural networks are proved to be almost sure exponentially stable. Numerical experiments are provided to illustrate the theoretical results which are derived in this paper. The computational efficiency of the balanced methods is demonstrated by numerical tests and compared to the Euler–Maruyama approximation scheme of the stochastic delay Hopfield neural networks. Furthermore, the obtained numerical results show that the balanced numerical methods of stochastic delay Hopfield neural networks are very efficient with the least error and have the best step size region for almost sure mean square stable.