Existence, Uniqueness and Exponential Stability of Periodic Solution for Discrete-Time Delayed BAM Neural Networks Based on Coincidence Degree Theory and Graph Theoretic Method

Manickam Iswarya,Chuangxia Huang,Jehad Alzabut,Jinde Cao,Grienggrai Rajchakit,Ramachandran Raja

doi:10.3390/math7111055

Abstract

In this work, a general class of discrete time bidirectional associative memory (BAM) neural networks (NNs) is investigated. In this model, discrete and continuously distributed time delays are taken into account. By utilizing this novel method, which incorporates the approach of Kirchhoff’s matrix tree theorem in graph theory, Continuation theorem in coincidence degree theory and Lyapunov function, we derive a few sufficient conditions to ensure the existence, uniqueness and exponential stability of the periodic solution of the considered model. At the end of this work, we give a numerical simulation that shows the effectiveness of this work.

Highlights

Differential and difference dynamic models have been intensively investigated because of their significance and applications in areas such as physics, mathematical biology and artificial neural networks [1,2,3,4,5,6,7,8,9,10,11,12]
Motivated by the aforementioned works, we investigated the discrete time Bidirectional associative memory neural networks (BAMNNs) with mixed time-varying delays which are exponentially stable and have a unique periodic solution
Together with the discrete and continuously distributed delay, the existence and periodic solution of discrete time BAMNNs is firstly proposed in the base of the graph theoretic approach, which generalizes and improves on the existing literature

Summary

Introduction

Differential and difference dynamic models have been intensively investigated because of their significance and applications in areas such as physics, mathematical biology and artificial neural networks [1,2,3,4,5,6,7,8,9,10,11,12]. The construction of Lyapunov functional for a large scale system is not an easy job To overcome this computational complexity, Li et al [53] proposed the novel graph-theoretic approach. There are few works on the exponential stability of periodic solution for discrete time delayed BAMNNs(DDBAMNNs). In this manuscript, together with the discrete and continuously distributed delay, the existence and periodic solution of discrete time BAMNNs is firstly proposed in the base of the graph theoretic approach, which generalizes and improves on the existing literature. At the end of this work, to show the exactness of this work, we present a numerical simulation

Preliminaries and Model Description

The Existence of Periodic Solution for DBAMNNs

Uniqueness and Exponential Stability of Periodic Solution

Illustrative Example

Conclusions