Restrictions and Stability of Time-Delayed Dynamical Networks

TL;DR

This paper investigates the stability of time-delayed dynamical networks, revealing conditions under which delays affect stability and introducing a new, efficient method for stability analysis, especially for neural networks.

Contribution

It introduces a novel procedure of dynamical network restrictions that improves stability estimates and simplifies analysis compared to previous methods.

Findings

Non-distributed delays preserve stability equivalence between delayed and non-delayed networks.

Distributed delays can alter the stability properties of the network.

The new restriction method enhances computational efficiency in stability analysis.

Abstract

This paper deals with the global stability of time-delayed dynamical networks. We show that for a time-delayed dynamical network with non-distributed delays the network and the corresponding non-delayed network are both either globally stable or unstable. We demonstrate that this may not be the case if the network's delays are distributed. The main tool in our analysis is a new procedure of dynamical network restrictions. This procedure is useful in that it allows for improved estimates of a dynamical network's global stability. Moreover, it is a computationally simpler and much more effective means of analyzing the stability of dynamical networks than the procedure of isospectral network expansions introduced in [Isospectral graph transformations, spectral equivalence, and global stability of dynamical networks. Nonlinearity, 25 (2012) 211-254]. The effectiveness of our approach is illustrated by applications to various classes of Cohen-Grossberg neural networks.