Exponential convergence of delayed dynamical systems

Global Exponential Stability for Complex-Valued Recurrent Neural Networks With Asynchronous Time Delays

In this paper, we investigate the global exponential stability for complex-valued recurrent neural networks with asynchronous time delays by decomposing complex-valued networks to real and imaginary parts and construct an equivalent real-valued system. The network model is described by a continuous-time equation. There are two main differences of this paper with previous works: 1) time delays can be asynchronous, i.e., delays between different nodes are different, which make our model more general and 2) we prove the exponential convergence directly, while the existence and uniqueness of the equilibrium point is just a direct consequence of the exponential convergence. Using three generalized norms, we present some sufficient conditions for the uniqueness and global exponential stability of the equilibrium point for delayed complex-valued neural networks. These conditions in our results are less restrictive because of our consideration of the excitatory and inhibitory effects between neurons; so previous works of other researchers can be extended. Finally, some numerical simulations are given to demonstrate the correctness of our obtained results.

Lag Synchronization of Memristor-Based Coupled Neural Networks via ω-Measure

This paper deals with the lag synchronization problem of memristor-based coupled neural networks with or without parameter mismatch using two different algorithms. Firstly, we consider the memristor-based neural networks with parameter mismatch, lag complete synchronization cannot be achieved due to parameter mismatch, the concept of lag quasi-synchronization is introduced. Based on the ω-measure method and generalized Halanay inequality, the error level is estimated, a new lag quasi-synchronization scheme is proposed to ensure that coupled memristor-based neural networks are in a state of lag synchronization with an error level. Secondly, by constructing Lyapunov functional and applying common Halanary inequality, several lag complete synchronization criteria for the memristor-based neural networks with parameter match are given, which are easy to verify. Finally, two examples are given to illustrate the effectiveness of the proposed lag quasi-synchronization or lag complete synchronization criteria, which well support theoretical results.

Global convergence of delayed dynamical systems

We discuss some delayed dynamical systems, investigating their stability and convergence in a critical case. To ensure the stability, the coefficients of the dynamical system must satisfy some inequalities. In most existing literatures, the restrictions on the coefficients are strict inequalities. The tough question is what will happen in the case (critical case) the strict inequalities are replaced by nonstrict inequalities (i.e., "<" is replaced by "=/<"). The purpose of the paper is to discuss this critical case and give an affirmative answer in the case that the activation functions are hyperbolic tangent.

Output convergence analysis for a class of delayed recurrent neural networks with time-varying inputs

This paper studies the output convergence of a class of recurrent neural networks with time-varying inputs. The model of the studied neural networks has different dynamic structure from that in the well known Hopfield model, it does not contain linear terms. Since different structures of differential equations usually result in quite different dynamic behaviors, the convergence of this model is quite different from that of Hopfield model. This class of neural networks has been found many successful applications in solving some optimization problems. Some sufficient conditions to guarantee output convergence of the networks are derived.

Some sufficient conditions for global exponential stability of delayed Hopfield neural networks

In this paper, we have derived some sufficient conditions for existence and uniqueness of equilibrium and global exponential stability in delayed Hopfield neural networks by using a different approach from the usually used one where the existence, uniqueness of equilibrium and stability are proved in two separate steps, rather we first prove global exponential convergence to 0 of the difference between any two solutions of the original neural networks, the existence and uniqueness of equilibrium is the direct results of this procedure. We obtain the conditions by suitable construction of Lyapunov functionals and estimation of derivates of the Lyapunov functionals by the well-known Young's inequality and Holder's inequality. The proposed conditions are related to p-norms of vector or matrix, p in [1, infinity] and thus unify and generalize some results in the literature.

Global convergence of delayed neural network systems

In this paper, without assuming the boundedness, strict monotonicity and differentiability of the activation functions, we utilize a new Lyapunov function to analyze the global convergence of a class of neural networks models with time delays. A new sufficient condition guaranteeing the existence, uniqueness and global exponential stability of the equilibrium point is derived. This stability criterion imposes constraints on the feedback matrices independently of the delay parameters. The result is compared with some previous works. Furthermore, the condition may be less restrictive in the case that the activation functions are hyperbolic tangent.

Global exponential stability of delayed Hopfield neural networks

In this paper, we discuss delayed Hopfield neural networks, investigating their global exponential stability. Sufficient conditions ensuring global exponential stability of delayed Hopfield neural networks are given.