Global stability and local stability of Hopfield neural networks with delays

Abstract

It is well known that Hopfield neural networks without delays exhibit no oscillations and possess global stability (i.e., all trajectories tend to some equilibrium). In the present paper we show that if the bound \ensuremath{\tau}\ensuremath{\beta}\ensuremath{\parallel}${\mathit{T}}_{2}$\ensuremath{\parallel}1 is satisfied, then a corresponding Hopfield neural network with delays \ensuremath{\tau}>0, interconnection matrix ${\mathit{T}}_{2}$ associated with delays, and gain of the neurons given by \ensuremath{\beta}, will exhibit similar qualitative properties as the original Hopfield neural network without delays (\ensuremath{\parallel}${\mathit{T}}_{2}$\ensuremath{\parallel} denotes the matrix norm induced by the Euclidean vector norm). Specifically, we show that if the above bound is satisfied, then a Hopfield neural network without delays and a corresponding Hopfield neural network with delays will have identical asymptotically stable equilibria, and both networks are globally stable. In addition to the above, we provide in the present paper an effective method of determining the asymptotic stability of an equilibrium of a Hopfield neural network with delays, assuming that the above bound is satisfied. Our results are consistent with the results reported by Marcus and Westervelt [Phys. Rev. A 39, 347 (1989)]. Specifically, the present results, all of which are obtained by rigorous proof, give support to these results, which are based on linearization arguments, numerical simulations, and experimental results.