Convergence and periodicity of solutions for a class of difference systems

Honghua Bin,Guang Zhang,Lihong Huang

doi:10.1155/ade/2006/70461

Abstract

A class of difference systems of artificial neural network with two neurons is considered. Using iterative technique, the sufficient conditions for convergence and periodicity of solutions are obtained in several cases.

Highlights

Consider the following difference system of the form: xn+1 = λxn + f yn, n = 0,1,2, . . . , yn+1 = λyn + f xn, where λ ∈ (0, 1) is a constant, for any a, b ∈ R, f : R → R is given by ⎧⎪⎨1, f (u) = ⎪⎩0, u ∈ [a, b], u ∈/ [a, b]. (1.1) (1.2)The system (1.1) can be viewed as the discrete version of the following two-neuron network model: dx dt = −αx + β f y [t] (1.3) dy dt
The aim of this paper is to investigate the convergence and periodicity of solutions for system (1.1) as f is of the digital nature (1.2), which describes the input-output relation of a neuron
It is easy to see that Theorems 2.5–2.9 and Propositions 2.3–2.12 are valid as a = −∞

Summary

Introduction

Consider the following difference system of the form: xn+1 = λxn + f yn , n = 0,1,2, . The system (1.1) can be viewed as the discrete version of the following two-neuron network model: dx dt. We may say that (1.1) includes the discrete version of an artificial neural network of two neurons with piecewise constant argument. Yuan et al [10] considered system (1.1), where the signal function f is of the following piecewise constant McCulloch-Pitts nonlinearity: f (u) = 1 if u ≤ σ, f (u) = −1 if u > σ, for some constant σ ∈ R. Let N denote the set of all nonnegative integers, and define N(m) = {m, m + 1, m + 2, .

Main results

Proofs of main results