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Global attractivity criteria for a discrete-time Hopfield neural network model with unbounded delays via singular M-matrices

José J. Oliveira, Ana Sofia Teixeira

TL;DR

Problem: derive global attractivity criteria for a discrete-time non-autonomous Hopfield neural network with infinite distributed delays and leakage-term delays. Approach: develop two $M$-matrix–based criteria—one for bounded activation functions requiring no invertibility, and a second for unbounded activations requiring a singular irreducible $M$-matrix—without Lyapunov functionals. Contributions: rigorous global attractivity results under (i) bounded activations with $\hat{\mathcal{M}}$ an $M$-matrix and (ii) unbounded activations with $\mathcal{M}^+$ singular irreducible $M$-matrix, supported by boundedness lemmas and $f^*,g^*$-based estimates, plus two numerical examples demonstrating applicability to non-autonomous, infinite-delay settings. Significance: broadens stability analysis for discrete-time Hopfield networks to include unbounded activation functions and singular $M$-matrices, extending prior work that focused on non-singular matrices and finite delays.

Abstract

In this work, we establish two global attractivity criteria for a multidimensional discrete-time non-autonomous Hopfield neural network model with infinite delays and delays in the leakage terms. The first criterion, which applies when the activation functions are bounded, is based on M-matrices that are not necessarily invertible. The second criterion, relevant for unbounded activation functions, requires that a related singular M-matrix be irreducible. We contrast our findings with existing results in the literature and present numerical simulations to illustrate the efficacy of the proposed criteria.

Global attractivity criteria for a discrete-time Hopfield neural network model with unbounded delays via singular M-matrices

TL;DR

Problem: derive global attractivity criteria for a discrete-time non-autonomous Hopfield neural network with infinite distributed delays and leakage-term delays. Approach: develop two -matrix–based criteria—one for bounded activation functions requiring no invertibility, and a second for unbounded activations requiring a singular irreducible -matrix—without Lyapunov functionals. Contributions: rigorous global attractivity results under (i) bounded activations with an -matrix and (ii) unbounded activations with singular irreducible -matrix, supported by boundedness lemmas and -based estimates, plus two numerical examples demonstrating applicability to non-autonomous, infinite-delay settings. Significance: broadens stability analysis for discrete-time Hopfield networks to include unbounded activation functions and singular -matrices, extending prior work that focused on non-singular matrices and finite delays.

Abstract

In this work, we establish two global attractivity criteria for a multidimensional discrete-time non-autonomous Hopfield neural network model with infinite delays and delays in the leakage terms. The first criterion, which applies when the activation functions are bounded, is based on M-matrices that are not necessarily invertible. The second criterion, relevant for unbounded activation functions, requires that a related singular M-matrix be irreducible. We contrast our findings with existing results in the literature and present numerical simulations to illustrate the efficacy of the proposed criteria.

Paper Structure

This paper contains 5 sections, 7 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Model Description
  3. Global attractiveness
  4. Numerical Examples
  5. Conclusions

Key Result

Theorem 2.1

Berman+Plemmons Let $A\in Z^{n\times n}$. The matrix $A$ is an $M-$matrix if and only if, for any $\overline{z}=(z_1,\ldots,z_n)\neq\overline{0}$ and taking $\overline{y}=(y_1,\ldots,y_n)=\left(A\overline{z}^T\right)^T$, there is $i\in[1,n]_\mathbb{Z}$ such that $z_i\neq0$ and $z_iy_i\geq0$.

Figures (2)

  • Figure 1: Plot of a solution $\overline{x}(m)=(x_1(m),x_2(m),x_3(m))$ of the numerical Example 4.1..
  • Figure 2: Plot of a solution $\overline{x}(m)=(x_1(m),x_2(m),x_3(m))$ of the numerical Example 4.2..

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 6 more