Discrete Time Delay Research Articles

Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse

In this letter we propose a class of linear fractional difference equations with discrete-time delay and impulse effects. The exact solutions are obtained by use of a discrete Mittag-Leffler function with delay and impulse. Besides, we provide comparison principle, stability results and numerical illustration.

Effect of cellular reservoirs and delays on the global dynamics of HIV

We investigate general HIV infection models with three types of infected cells: latently infected cells, long-lived productively infected cells, and short-lived productively infected cells. We incorporate three discrete or distributed time delays into the models. Moreover, we consider the effect of humoral immunity on the dynamical behavior of the HIV. The HIV-target incidence rate, production/proliferation, and removal rates of the cells and HIV are represented by general nonlinear functions. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive two threshold parameters which fully determine the existence and stability of the three steady states of the model. Using Lyapunov functionals, we establish the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations.

Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays

In this paper, a class of complex-valued neural network model with discrete and distributed delays is proposed. Regarding the discrete time delay as the bifurcating parameter, the problem of Hopf bifurcation in the newly-proposed complex-valued neural network model is investigated under the assumption that the activation function can be separated into its real and imaginary parts. Based on the normal form theory and center manifold theorem, some sufficient conditions which determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established. Finally, a numerical example is given to illustrate the validity of the theoretical results.

Global exponential synchronization of nonautonomous recurrent neural networks with time delays on time scales

This paper is concerned on the global exponential synchronization in timescale sense for a class of nonautonomous recurrent neural networks (NRNNs) with discrete-time delays on time scales. Firstly, a timescale-type comparison result is given based on the induction principle of time scales. Then by the constructed comparison lemma, the theory of time scales and analytical techniques, several synchronization criteria for the driven and response NRNNs are obtained. Moreover, several examples are given to show the effectiveness and validity of the main results. The obtained synchronization criteria improve or extend some existing ones in the literature.

Stability and Hopf Bifurcation of a Three-Neuron Network with Multiple Discrete and Distributed Delays

In this paper, a class of three-neuron network with discrete and distributed delays is introduced. We first give a detailed Hopf bifurcation analysis for the proposed network. Choosing the discrete time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Moreover, by using the normal form theory and center manifold theorem, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, numerical simulations are presented to demonstrate the effectiveness of our theoretical results.

Analysis of an economic model of Kaldor type modified with two discrete time delays

Analysis of an economic model of Kaldor type modified with two discrete time delays

Analysis of latent CHIKV dynamics models with general incidence rate and time delays

ABSTRACTIn this paper, we study the stability analysis of latent Chikungunya virus (CHIKV) dynamics models. The incidence rate between the CHIKV and the uninfected monocytes is modelled by a general nonlinear function which satisfies a set of conditions. The model is incorporated by intracellular discrete or distributed time delays. Using the method of Lyapunov function, we established the global stability of the steady states of the models. The theoretical results are confirmed by numerical simulations.

Nonlinear dynamic analysis on maglev train system with flexible guideway and double time-delay feedback control

In this paper, the dynamic behavior of time-delayed feedback control for maglev train system with double discrete time delays is considered with flexible guideway. Considering the maglev guideway as Beroulli-Euler beam, the mathematical model of maglev system with flexible guideway is constructed. The time delay of the two state feedback signals in the maglev system occurs simultaneously, and the values are different. The present treatment method only considers one single feedback delay, which are insufficiency. Thus, the Hopf bifurcation with double time-delay feedback of maglev train running on the flexible guideway is analyzed considering time-delayed position feedback signal τ1 and velocity feedback signal τ2. A novel method is presented to develop the double-parametric Hopf bifurcation diagram in relation to τ1 and τ2. Sufficient numerical simulations are provided to illustrate the complex dynamical behavior of the discrete delays τ1 and τ2 for maglev system and we verify the obtained theoretical analysis. Finally, the field experiments are carried out to validate the effectiveness of the Hopf bifurcation analytical method preliminarily.

Global Asymptotic Stability of a Functional Differential Model with Time Delay of an Anaerobic Biodegradation Process

We study a nonlinear functional differential model of an anaerobicdigestion process of wastewater treatment with biogas production. Themodel equations of biomass include two different discrete time delays. Amathematical analysis of the model is completed including existence andlocal stability of nontrivial equilibrium points, existence and boundednessof the model solutions as well as global stabilizability towards an admissibleequilibrium point. We propose and apply a numerical extremum seekingalgorithm for maximizing the biogas flow rate in real time. Numerical simulation results are also included.ACM Computing Classification System (1998): D.2.6, G.1.10, J.2.

Impulsive Cohen–Grossberg BAM neural networks with mixed time-delays: An exponential stability analysis issue

Abstract This paper investigates, the globally exponential stability analysis problem for a class of markovian jumping Cohen–Grossberg BAM-type neural networks (CGBAMNNs) with mixed time delays and impulsive effects. Here the jumping parameters are considered, which are governed by a markov process with discrete & finite state space. The mixed time delays carries both discrete time-varying and distributed delays, which means the lower and upper bounds of discrete time delays are available. By fabricating an appropriate Lyapunov–Krasovskii functional (LKF), some new sufficient conditions are obtained in terms of linear matrix inequalities (LMIs) to guarantee the globally exponential stability for the labeled neural networks. The obtained conditions are expressed in terms of LMIs whose feasibility can be checked easily by MATLAB LMI control toolbox. Furthermore, we have collated our effort with foregoing one in the available literatures and showed that it is less conserved. Finally, three numerical examples with their simulative reactions are provided to demonstrate the viability of the notional outcomes.

Traveling wave phenomena of [formula omitted]-dimensional diffusive predator–prey systems

In this work, we deal with the n-dimensional reaction-diffusive predator–prey systems with the discrete time delay, which have numerous applications in biology and ecology. By applying an asymptotic analysis, the Schauder’s fixed point theorem as well as the upper and lower solution method, we investigate the existence, non-existence, exponentially asymptotic stability and asymptotic behaviors of traveling wave solutions with the appropriate small delay when the wave speed c varies. Moreover, we derive the minimal wave speed c∗ based on the well-established properties of the Fisher equation, and prove the existence of traveling wave solutions when the wave speed c≥c∗, which provides a sharp contrast to the case of c<c∗. The obtained results extend the existing ones in the literature to arbitrary finite dimensional systems with the discrete time delay and demonstrate asymptotic behaviors of traveling wave solutions.

Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval.

Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life style. Both diabetic population and health expenses grow rapidly according to American Diabetes Association. Detecting the potential onset of T2DM is an essential focal point in the research of diabetes mellitus. The intravenous glucose tolerance test (IVGTT) is an effective protocol to determine the insulin sensitivity, glucose effectiveness, and pancreatic β-cell functionality, through the analysis and parameter estimation of a proper differential equation model. Delay differential equations have been used to study the complex physiological phenomena including the glucose and insulin regulations. In this paper, we propose a novel approach to model the time delay in IVGTT modeling. This novel approach uses two parameters to simulate not only both discrete time delay and distributed time delay in the past interval, but also the time delay distributed in a past sub-interval. Normally, larger time delay, either a discrete or a distributed delay, will destabilize the system. However, we find that time delay over a sub-interval might not. We present analytically some basic model properties, which are desirable biologically and mathematically. We show that this relatively simple model provides good fit to fluctuating patient data sets and reveals some intriguing dynamics. Moreover, our numerical simulation results indicate that our model may remove the defect in well known Minimal Model, which often overestimates the glucose effectiveness index.

Parameter identification of nonlinear multirate time-delay system with uncertain output delays

The joint parameter and time-delay estimation problems for a class of nonlinear multirate time-delay system with uncertain output delays are addressed in this paper. The practical process typically has time-delay properties and the process data are often multirate, sampled with output data inevitably corrupted by uncertain delays. The linear parameter varying (LPV) finite impulse response (FIR) multirate time-delay model is initially built to describe the considered system. The problems of over-parameterization and the existence of both continuous model parameters and discrete time-delays have made the conventional maximum likelihood difficult to solve the considered problems. In order to handle these problems, the joint parameter and time-delay estimation for the LPV FIR multirate time-delay model are formulated in the expectation-maximization scheme, and the algorithm to estimate the model parameters and time-delays is derived, simultaneously based on multirate process data. The efficacy of the proposed method is verified through a numerical simulation and a practical chemical plant.

Input-to-State Stability of Discrete-Time Delay Systems with Delayed Impulses

This paper investigates the input-to-state stability (ISS) of discrete-time delay systems with delayed impulses. By employing Lyapunov functions together with Razumikhin technique, a number of ISS criteria are obtained. It is shown that if the original impulse-free system has no ISS property, it can become input-to-state stable by adding certain impulsive controls which are stabilizing. Correspondingly, the original system without impulses can retain its ISS property with appropriate impulsive perturbations which are destabilizing. Then, robust ISS property for a class of uncertain discrete-time delay systems with delayed impulses is considered. Finally, some numerical examples are given to illustrate the effectiveness and the superiority of the results.

Teaching multi-step requesting to children with autism spectrum disorder using systematic instruction and a speech-generating device

The purpose of this study was to examine the effectiveness of an iPad-based speech-generating device (SGD) and an intervention package in teaching multi-step requesting to children with autism spectrum disorder (ASD). The intervention package comprised discrete trial teaching, time delay, graduated guidance, and reinforcement. Social validity data were also collected from parents and teachers. Three male participants with ASD, aged 4–5 years, participated in the study, which was conducted using a multiple-probe-across-participants design. Findings of the study showed that the SGD and the intervention package were effective in teaching multi-step requesting to all participants. Furthermore, the target skill was maintained and generalized to different materials and individuals. Social validity findings indicated that opinions of the mothers and teachers were positive. On the other hand, the father of one participant stated that he was concerned with possible negative effects of using tablet computers. The findings are discussed with regard to the parents’ opinions, and implications for practice and research.

Control strategies of avian influenza pandemic model with time delay

A mathematical model for avian influenza with optimal control strategies is presented as a system of discrete time delay differential equations (DDEs) and its important mathematical features are analyzed. In alignment to manage this, we develop an optimally controlled pandemic model of avian influenza and insert a time delay with exponential factor. Then we apply two controlled functions in the form of biosecurity of poultry and the education campaign against avian influenza to control the disperse of the disease. Our optimal control strategies will minimize the number of contaminated humans and contaminated birds. We also derive the basic reproduction number to examine the dynamical behavior of the model and demonstrate the existence of the controlled system. For the justification of our work, we present numerical simulations.

Robust stability analysis of impulsive complex-valued neural networks with time delays and parameter uncertainties

The present study considers the robust stability for impulsive complex-valued neural networks (CVNNs) with discrete time delays. By applying the homeomorphic mapping theorem and some inequalities in a complex domain, some sufficient conditions are obtained to prove the existence and uniqueness of the equilibrium for the CVNNs. By constructing appropriate Lyapunov-Krasovskii functionals and employing the complex-valued matrix inequality skills, the study finds the conditions to guarantee its global robust stability. A numerical simulation illustrates the correctness of the proposed theoretical results.

Modeling the Control of Algal Bloom in a Lake by Applying Some External Efforts with Time Delay

One of the crucial issues in ecology is related to algal bloom in lakes. Vast amount of efforts has been channeled to combat this problem, but for successful implementation of any algae eradication effort, their impact on the dynamics of water body must be studied explicitly. In view of this, we propose and analyze a mathematical model to assess the effect of applying efforts for removal of algae and detritus from a lake on the occurrence of algal bloom. The system is governed by the interactions of four variables, namely, concentration of nutrients, density of algae, density of detritus and efforts applied to remove algae as well as detritus from the lake. The biologically feasible equilibria and their stability properties are analyzed and discussed. Considering the growth rate of applied efforts as time dependent, we also obtain the optimal control strategy to minimize both the algae and the associated costs. The optimality system is derived and solved numerically. Further, to make the system more realistic, two discrete time delays; one in conversion of detritus into nutrients and other in applying the control efforts are considered. We study various cases for the time delays and show that in general the system develops limit cycle oscillation through a Hopf-bifurcation for increasing time delays. However, we observe that in some cases, increase in time delays may diminish the oscillations. Our results indicate that high concentration of nutrient intensifies the occurrence of algal bloom and in that case applied efforts play a pivotal role in reducing the bloom.

A mathematical model for the conservation of forestry resources with two discrete time delays

Forests are very important for the life on the planet earth. Now-a-days, depletion of forestry resources is a serious problem across the world and there is a need to make some efforts to conserve them. For this, in the present study, we have formulated a nonlinear mathematical model to assess the effect of applied technological efforts on the conservation of forestry resources with two discrete time delays. The first time delay is involved in applying technological efforts whereas the second one is responsible for the visibility of applied technological efforts on the growth of forestry resources. The conditions for local stability are obtained and it is shown that periodic solutions arise through Hopf-bifurcation as the time delay crosses some threshold value. Further, it is noted that the stable equilibrium may become unstable due to the increase in the value of second delay parameter. Numerical simulation is also performed to support the analytically obtained results.

Single-shot Monitoring of Ultrafast Processes via X-ray Streaking at a Free Electron Laser

The advent of x-ray free electron lasers has extended the unique capabilities of resonant x-ray spectroscopy techniques to ultrafast time scales. Here, we report on a novel experimental method that allows retrieving with a single x-ray pulse the time evolution of an ultrafast process, not only at a few discrete time delays, but continuously over an extended time window. We used a single x-ray pulse to resolve the laser-induced ultrafast demagnetisation dynamics in a thin cobalt film over a time window of about 1.6 ps with an excellent signal to noise ratio. From one representative single shot measurement we extract a spin relaxation time of (130 ± 30) fs with an average value, based on 193 single shot events of (113 ± 20) fs. These results are limited by the achieved experimental time resolution of 120 fs, and both values are in excellent agreement with previous results and theoretical modelling. More generally, this new experimental approach to ultrafast x-ray spectroscopy paves the way to the study of non-repetitive processes that cannot be investigated using traditional repetitive pump-probe schemes.