Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

TL;DR

This paper establishes new stability criteria for general delay difference equations and applies these results to neural network models with delays, proving the existence and stability of periodic solutions.

Contribution

It introduces a novel global exponential stability criterion for multidimensional delay difference equations and demonstrates their application to neural networks with delayed leakage terms.

Findings

Established a new stability criterion for delay difference equations.

Proved existence of periodic solutions using Poincaré map.

Derived stability conditions for neural networks including Hopfield and BAM models.

Abstract

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincar\'e map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory(BAM).