Exponential periodicity and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays

Abstract

In this paper, the exponential periodicity and stability of neural networks with Lipschitz continuous activation functions are investigated, without assuming the boundedness of the activation functions and the differentiability of time-varying delays, as needed in most other papers. The neural networks contain reaction-diffusion terms and both variable and unbounded delays. Some sufficient conditions ensuring the existence and uniqueness of periodic solution and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays are obtained by analytic methods and inequality technique. Furthermore, the exponential converging index is also estimated. The methods, which does not make use of Lyapunov functional, is simple and valid for the periodicity and stability analysis of neural networks with variable and/or unbounded delays. The results extend some previous results. Two examples are given to show the effectiveness of the obtained results.

Exponential stability of reaction-diffusion generalized Cohen-Grossberg neural networks with both variable and distributed delays

“…It is not difficult to discover that Theorem 1 in [29] is involved in our Corollary 2. However, since there exist reaction-diffusion terms Corollary 4 in this paper is not as good as Theorem 1 in [27].…”

“…A more appropriate and ideal way is to incorporate finite delays and infinite delays, e.g.,Refs. [27]- [29]. However, strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields.…”

“…So we must consider that the activations vary in space as well as in time. In [4]- [7], [14]- [17], [23], [29,30], [34,36], the authors have considered the stability of neural networks with diffusion terms, which are expressed by partial differential equations. It is also common to consider the diffusion effect in biological systems such as immigration.…”

“…Although the use of finite delays in models with delayed feedback provides a good approximation to simple circuits consisting of a small number of neurons, neural networks usually should have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there will be a distributed of propagation delays in finite or/and infinite time (see [6], [13]- [16], [25]- [29], [38]). In the case, the signal propagation is no longer instantaneous and cannot be modeled with finite delays or infinite delays.…”

Delay-independent stability of stochastic reaction–diffusion neural networks with Dirichlet boundary conditions

“…Hence, it is essential to consider the state variables varying with time and space. The neural networks with diffusion terms can commonly be expressed by partial differential equations [10][11][12][13][14][15] have considered the stability of neural networks with diffusion terms, in which boundary conditions are all the Neumann boundary conditions. The neural networks model with Dirichlet boundary conditions has been considered in [16,21], but it concentrated on deterministic systems and did not take random perturbation into consideration.…”