Random attractors of a stochastic Hopfield neural network model with delays

TL;DR

This paper investigates the long-term behavior of a stochastic Hopfield neural network with delays by establishing the existence of random attractors and analyzing their properties through rigorous mathematical methods and numerical simulations.

Contribution

It proves the existence of random attractors for stochastic delayed Hopfield neural networks and characterizes their structure and stability, which was not previously established.

Findings

Existence of a continuous, compact trajectory field for the stochastic delayed HNNM.

Existence of a random absorbing set and attractors under certain conditions.

Numerical simulations confirm the theoretical results.

Abstract

The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for $t>\tau$ (where $\tau$ is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained by a careful uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is proved and hence the existence of random attractors is obtained. Moreover, sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are given. Numerical simulations are also conducted at last to illustrate the effectiveness of the established results.