Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations

This paper investigates the dynamics of a class of recurrent neural networks where the neural activations are modeled by discontinuous functions. Without presuming the boundedness of activation functions, the sufficient conditions to ensure the existence, uniqueness, global exponential stability and global convergence of state equilibrium point and output equilibrium point are derived, respectively. Furthermore, under certain conditions we prove that the system is convergent globally in finite time. The analysis in the paper is based on the properties of M-matrix, Lyapunov-like approach, and the theories of differential equations with discontinuous right-hand side as introduced by Filippov. The obtained results extend previous works on global stability of recurrent neural networks with not only Lipschitz continuous but also discontinuous neural activation functions.     

Almost periodic solutions of shunting inhibitory cellular neural networks on time scales

In this paper shunting inhibitory cellular neural networks (SICNNs) with time-varying and continuously distributed delays are considered on time scale T. Without assuming the global Lipschitz conditions of activation functions, some new sufficient conditions for the existence and asymptotic stability of the almost periodic solutions are established on time scales. Two numerical examples are given to illustrate our feasible results.     

变时滞BAM神经网络的全局指数稳定性

研究一类变时滞BAM神经网络平衡点的全局指数稳定性问题.在不要求激励函数全局Lipschitz条件下,通过构造合适的Lyapunov泛函,并结合Young不等式,得到了BAM神经网络模型在一定条件下全局指数稳定的一些充分条件,推广和改进了前人的相关结论,为综合设计指数稳定的时滞BAM神经网络提供了依据.     

Stability analysis of impulsive Cohen–Grossberg neural networks with distributed delays and reaction–diffusion terms

In this paper, we investigate a class of impulsive Cohen–Grossberg neural networks with distributed delays and reaction–diffusion terms. By establishing an integro-differential inequality with impulsive initial conditions and applying M-matrix theory, we find some sufficient conditions ensuring the existence, uniqueness, global exponential stability and global robust exponential stability of equilibrium point for impulsive Cohen–Grossberg neural networks with distributed delays and reaction–diffusion terms. An example is given to illustrate the results obtained here.     

Dynamical behavior of delayed Hopfield neural networks with discontinuous activations

In this paper, a general class of neural networks with arbitrary constant delays is studied, whose neuron activations are discontinuous and may be unbounded or nonmonotonic. Based on the Leray–Schauder alternative principle and generalized Lyapunov approach, conditions are given under which there is a unique equilibrium of the neural network, which is globally asymptotically stable. Moreover, the existence and global asymptotic stability of periodic solutions are derived, where the neuron inputs are periodic. The obtained results extend previous works not only on delayed neural networks with Lipschitz continuous neuron activations, but also on delayed neural networks with discontinuous neuron activations.     

Almost periodicity for a class of delayed Cohen–Grossberg neural networks with discontinuous activations

The objective of this paper is to investigate the dynamics of a class of delayed Cohen–Grossberg neural networks with discontinuous neuron activations. By means of retarded differential inclusions, we obtain a result on the local existence of solutions, which improves the previous related results for delayed neural networks. It is shown that an M-matrix condition satisfied by the neuron interconnections, can guarantee not only the existence and uniqueness of an almost periodic solution, but also its global exponential stability. It is also shown that the M-matrix condition ensures that all solutions of the system display a common asymptotic behavior. In this paper, we prove that the existence interval of the almost periodic solution is (?∞, +∞), whereas the existence interval is only proved to be [0, +∞) in most of the literature. As special cases, we derive the results of existence, uniqueness and global exponential stability of a periodic solution for delayed neural networks with periodic coefficients, as well as the similar results of an equilibrium for the systems with constant coefficients. To the author’s knowledge, the results in this paper are the only available results on almost periodicity for Cohen–Grossberg neural networks with discontinuous activations and delays.     

一类变时滞神经网络的全局指数稳定性

本文研究一类变时滞神经网络平衡点的全局指数稳定性.在不要求激活函数全局Lipschitz条件下,利用Lyapunov函数方法,并结合Young不等式和Halanay时滞微分不等式,得到了系统全局指数稳定的充分条件.文末,一个数值例子用以说明本文结果的有效性.     

SCALE-TYPE STABILITY FOR NEURAL NETWORKS WITH UNBOUNDED TIME-VARYING DELAYS

This paper studies scale-type stability for neural networks with unbounded time-varying delays and Lipschitz continuous activation functions. Several sufficient conditions for the global exponential stability and global asymptotic stability of such neural networks on time scales are derived. The new results can extend the existing relevant stability results in the previous literatures to cover some general neural networks.     

Periodic solutions of high-order Cohen–Grossberg neural networks with distributed delays

A class of high-order Cohen–Grossberg neural networks with distributed delays is investigated in this paper. Sufficient conditions to guarantee the uniqueness and global exponential stability of periodic solutions of such networks are established by using suitable Lypunov function and the properties of M-matrix. The results in this paper improve the earlier publications.     

Exponential stability of fuzzy Cohen–Grossberg neural networks with time delays and impulsive effects

In this paper, we investigate a class of fuzzy Cohen-Grossberg neural networks with time delays and impulsive effects. By employing an inequality technique, we find sufficient conditions for the existence, uniqueness, global exponential stability of the equilibrium without using the M-matrix theory. An example is given to illustrate the effectiveness of the obtained results.     

PERIODIC SOLUTIONS TO FUZZY CELLULAR NEURAL NETWORKS WITH DISTRIBUTED DELAYS

In this paper, a class of fuzzy cellular neural networks with distributed delays is discussed. By employing fixed point theorem and inequality techniques, some sufficient conditions are obtained to ensure the existence and global exponential stability of periodic solutions to the systems. Without assuming the global Lipschitz conditions of activation functions, our results are novel and reduce the limitation of previous known results. Moreover, an example is given to illustrate the effectiveness of our resu...     

Lasalle method and general decay stability of stochastic neural networks with mixed delays

This paper investigates the general decay pathwise stability conditions on a class of stochastic neural networks with mixed delays by applying Lasalle method. The mixed time delays comprise both time-varying delays and infinite distributed delays. The contributions are as follows: (1)?we extend the Lasalle-type theorem to cover stochastic differential equations with mixed delays; (2)?based on the stochastic Lasalle theorem and the M-matrix theory, new criteria of general decay stability, which includes the almost surely exponential stability and the almost surely polynomial stability and the partial stability, for neural networks with mixed delays are established. As an application of our results, this paper also considers a two-dimensional delayed stochastic neural networks model.     

Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author’s earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.     

New results for global stability of a class of neutral-type neural systems with time delays

This paper studies the global convergence properties of a class of neutral-type neural networks with discrete time delays. This class of neutral systems includes Cohen–Grossberg neural networks, Hopfield neural networks and cellular neural networks. Based on the Lyapunov stability theorems, some delay independent sufficient conditions for the global asymptotic stability of the equilibrium point for this class of neutral-type systems are derived. It is shown that the results presented in this paper for neutral-type delayed neural networks are the generalization of a recently reported stability result. A numerical example is also given to demonstrate the applicability of our proposed stability criteria.     

Positive invariant sets and global exponential attractive sets of a class of neural networks with unbounded time-delays

In this paper, we study the positive invariant sets and global exponential attractive sets for a class of neural networks with unbounded time-delays. Based on the assumption for the activation function satisfying the global Lipschitz condition, several algebraic criterions for the aforementioned sets are obtained by constructing proper Lyapunov functions and employing Young inequality. Finally, examples are given and analyzed to demonstrate our results.     

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

In this paper, we investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.     

New results of almost periodic solutions for recurrent neural networks

In this paper recurrent neural networks with time-varying delays and continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, some sufficient conditions for the existence and local exponential stability of the almost periodic solutions are established, which are new and complement previously known results.     

Global asymptotic stability of a class of nonautonomous integro-differential systems and applications

In this paper, we study the global asymptotic stability of a class of nonautonomous integro-differential systems. By constructing suitable Lyapunov functionals, we establish new and explicit criteria for the global asymptotic stability in the sense of Definition 2.1. In the autonomous case, we discuss the global asymptotic stability of a unique equilibrium of the system, and in the case of periodic system, we establish sufficient criteria for existence, uniqueness and global asymptotic stability of a periodic solution. Also explored are applications of our main results to some biological and neural network models. The examples show that our criteria are more general and easily applicable, and improve and generalize some existing results.     

Stability in impulsive Cohen-Grossberg-type BAM neural networks with distributed delays

In this paper, a class of impulsive Cohen-Grossberg-type bi-directional associative memory (BAM) neural networks with distributed delays is investigated. By establishing an integro-differential inequality with impulsive initial conditions and employing the homeomorphism theory, the M-matrix theory and inequality technique, some new general sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive Cohen-Grossberg-type BAM neural networks with distributed delays are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on the system parameters and impulsive disturbed intension. An example is given to show the effectiveness of the results obtained here.