Multiple stability and instability of Cohen\u2013Grossberg neural network with unbounded time-varying delays

Yanheng Chen,Songfang Jia

doi:10.1186/s13660-019-2129-0

Abstract

In this paper, multiple stability and instability of Cohen–Grossberg neural network with unbounded time-varying delays are studied. Based on the geometrical configuration of activation functions and some rigorous mathematical analysis, some algebraic criteria are proposed to guarantee coexistence of multiple stable equilibrium points and multiple unstable equilibrium points in the model. Moreover, using the partition space method, we prove that the discussed model has at least 3^{n} equilibrium points, 2^{n} of them are locally μ-stable and others are unstable. Finally, the numerical example and its simulation show the effectiveness of the proposed results.

Highlights

The Cohen–Grossberg neural network model, proposed by Cohen and Grossberg in 1983 [1], has been attracting much attention because of its wide application in various engineering fields and because of it being highly inclusive of other neural networks such as Hopfield neural network, cellular neural network, recurrent neural network, and so on
From the references mentioned above, we find that the multistability of Cohen–Grossberg neural networks with unbounded time-varying delays is a challenging problem
Motivated by the challenging problem, we investigate the multistability of a Cohen– Grossberg neural network with unbounded time-varying delay and nondecreasing activation functions in this paper and prove that the considered model has 3n equilibrium points, and 2n of them are locally μ-stable, the remaining ones are unstable

Summary

Introduction

The Cohen–Grossberg neural network model, proposed by Cohen and Grossberg in 1983 [1], has been attracting much attention because of its wide application in various engineering fields and because of it being highly inclusive of other neural networks such as Hopfield neural network, cellular neural network, recurrent neural network, and so on. It is necessary that there exist multiple stable equilibrium points for neural networks. By using the partition space method, [41] proved that neural networks with unbounded time-varying delays could exhibit at least 3n equilibrium points, 2n of them are locally μ-stable and others are unstable.

Results

Conclusion