Finite-time uniform stability of functional differential equations with applications in network synchronization control

In this paper, we investigate finite-time uniform stability of functional differential equations with applications in network synchronization control. First, a Razumikhin-type theorem is derived to ensure finite-time uniform stability of functional differential equations. Based on the theoretical results, finite-time uniform synchronization is proposed for a class of delayed neural networks and delayed complex dynamical networks by designing nontrivial and simple control strategies and some novel criteria are established. Especially, a feasible region of the control parameters for each neuron is derived for the realization of finite-time uniform synchronization of the addressed neural networks, which provide a great convenience for the application of the theoretical results. Finally, two numerical examples with numerical simulations are provided to show the effectiveness and feasibility of the theoretical results.     

The oscillation of perturbed functional differential equations

We provide new oscillation criteria for the perturbed functional differential equations. This solves some open problems of [1]. An application to an equation arising in nonlinear neural networks is illustrated.     

Global exponential stability for a class of retarded functional differential equations with applications in neural networks

In this paper, the global exponential stability and asymptotic stability of retarded functional differential equations with multiple time-varying delays are studied by employing several Lyapunov functionals. A number of sufficient conditions for these types of stability are presented. Our results show that these conditions are milder and more general than previously known criteria, and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Furthermore, the results obtained for neural networks with time-varying delays do not assume symmetry of the connection matrix.     

Symmetric functional differential equations and neural networks with memory

We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

     

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.     

Encoding and decoding of analog signals with a population of neurons

In this paper, we discuss how an analog signal can be encoded using biophysically realistic neural networks. Using the activity curve of a single neuron, we argue that the activities can be pooled over a population so that the weighted sum of the activities approximate a given function. Since the activities of neurons are not available as a variable, we propose to generate them in real time by a suitable low-pass filter. Using the proposed scheme, we demonstrate how simple ordinary differential equations can be solved. In effect, the ordinary differential equations are solved by dynamically updating the activities of the neurons. In an actual biological neural network, the activities of the cells are not obtained by a low-pass filter. They are integrated in the network by a suitable synaptic input. A new optimization algorithm for finding a set of optimal synaptic weights has been proposed and successfully implemented using a software package GENESIS. The difference between biological neural networks and artificial neural networks is discussed in somewhat greater details. The important concepts are illustrated by implementing a memory and by solving a periodic ordinary differential equation, the Van der Pol oscillator.     

Implicit difference methods for evolution functional differential equations

A general theory of implicit difference schemes for nonlinear functional differential equations with initial boundary conditions is presented. A theorem on error estimates of approximate solutions for implicit functional difference equations of the Volterra type with an unknown function of several variables is given. This general result is employed to investigate the stability of implicit difference schemes generated by first-order partial differential functional equations and by parabolic problems. A comparison technique with nonlinear estimates of the Perron type for given functions with respect to the functional variable is used.     

On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays

This paper considers the impulsive functional differential equations with infinite delays or finite delays. Some new sufficient conditions are obtained to guarantee the global exponential stability by employing the improved Razumikhin technique and Lyapunov functions. The result extends and improves some recent works. Moreover, the obtained Razumikhin condition is very simple and effective to implement in real problems and it is helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation. Finally, a numerical example and its simulation is given to show the effectiveness of the obtained result in this paper.     

Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays

In this paper, we formulate and investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, the viability and dissipativity of solutions for functional differential inclusions and memristive BAM neural networks can be guaranteed by the matrix measure approach and generalized Halanay inequalities. Then, a new method involving the application of set-valued version of Krasnoselskii’ fixed point theorem in a cone is successfully employed to derive the existence of the positive periodic solution. The dynamic analysis in this paper utilizes the theory of set-valued maps and functional differential equations with discontinuous right-hand sides of Filippov type. The obtained results extend and improve some previous works on conventional BAM neural networks. Finally, numerical examples are given to demonstrate the theoretical results via computer simulations.     

Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

Numerical method of lines for parabolic functional differential equations

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given.     

Functional differential inclusions generated by functional differential equations with discontinuities

Given a functional differential equation with a discontinuity, a construction of its extension in the shape of a functional differential inclusion is offered. This construction can be regarded as a generalization of the famous Filippov approach to study ordinary differential equations with discontinuities. Some basic properties of the solutions of the introduced functional differential inclusions are studied. The developed approach is applied to analysis of gene regulatory networks with general delays.     

Tidal level forecasting using functional and sequential learning neural networks

Prediction of tides is very much essential for human activities and to reduce the construction cost in marine environment. This paper presents two methods (1) an application of the functional networks (FN) and (2) sequential learning neural network (SLNN) procedures for the accurate prediction of tides using very short-term observation. This functional network model predicts the time series data of hourly tides directly while using an efficient learning process by minimizing the error based on the observed data for 30 days. Using the functional network, a very simple equation in the form of finite difference equation using the tidal levels at two previous time steps is arrived at. Sequential learning neural network uses one hidden neuron to predict the current tidal level using the previous four levels quite accurately. Hourly tidal data measured at Taichung harbor and Mirtuor coast along the Taiwan coastal region have been used for testing the functional network and sequential neural network model. Results show that the hourly data on tides for even a month can be predicted efficiently with a very high correlation coefficient.     

Neural network solution of pantograph type differential equations

We investigate the approximate solution of pantograph type functional differential equations using neural networks. The methodology is based on the ideas of Lagaris et al, and itis applied to various problems with a proportional delay term subject to initial or boundary conditions. The proposed methodology proves to be very efficient.     

Nonlinear oscillation of fourth order functional differential equations

Summary In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) ₂ ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations. Entrata in Redazione il 5 settembre 1977.     

Anti-synchronization control of a class of memristive recurrent neural networks

In this paper, we formulate and investigate a class of memristive recurrent neural networks. Two different types of anti-synchronization algorithms are derived to achieve the exponential anti-synchronization of the coupled systems based on drive–response concept, differential inclusions theory and Lyapunov functional method. The proposed anti-synchronization algorithms are simple and can be easily realized. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. The obtained results extend some previous works on conventional recurrent neural networks.     

Existence and global exponential stability of almost periodic solutions for BAM neural networks with distributed leakage delays on time scales

In this paper, we deal with a class of BAM neural networks with distributed leakage delays on time scales. Some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such class of BAM neural networks are obtained by applying the exponential dichotomy of linear differential equations, Lapunov functional method and contraction mapping principle. An example is given to illustrate the effectiveness of the theoretical predictions. The obtained results in this paper are completely new and complement the previously known publications.     

Synchronization of neural networks based on parameter identification and via output or state coupling

For neural networks with all the parameters unknown, we focus on the global robust synchronization between two coupled neural networks with time-varying delay that are linearly and unidirectionally coupled. First, we use Lyapunov functionals to establish general theoretical conditions for designing the coupling matrix. Neither symmetry nor negative (positive) definiteness of the coupling matrix are required; under less restrictive conditions, the two coupled chaotic neural networks can achieve global robust synchronization regardless of their initial states. Second, by employing the invariance principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the robust synchronization of almost all kinds of coupled neural networks with time-varying delay based on the parameter identification of uncertain delayed neural networks. Finally, numerical simulations validate the effectiveness and feasibility of the proposed technique.     

Periodic solutions of some nonlinear autonomous functional differential equations

Summary We develop here some new fixed point theorems and apply them to the question of existence of nontrivial periodic solutions of nonlinear, autonomous functional differential equations. We prove that the standard results of G. S. Jones and R. B. Grafton can be obtained by our methods, and we prove periodicity results for some equations, for instance a neutral functional differential equation, which appear inaccessible by previous techniques. Partially supported by NSFGP 20228 and a Rutgers Research Council Faculty Fellowship. Entrata in Redazione il 10 gennaio 1973.     

Robust synchronization of delayed neural networks based on adaptive control and parameters identification

This paper investigates synchronization dynamics of delayed neural networks with all the parameters unknown. By combining the adaptive control and linear feedback with the updated law, some simple yet generic criteria for determining the robust synchronization based on the parameters identification of uncertain chaotic delayed neural networks are derived by using the invariance principle of functional differential equations. It is shown that the approaches developed here further extend the ideas and techniques presented in recent literature, and they are also simple to implement in practice. Furthermore, the theoretical results are applied to a typical chaotic delayed Hopfied neural networks, and numerical simulation also demonstrate the effectiveness and feasibility of the proposed technique.