Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.     

A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations

We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used.     

A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey

In this paper, we formulate a robust prey-dependent consumption predator-prey model with a delay of digestion and impulsive perturbation on the prey. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘predator-eradication’ periodic solution and show that the ‘predator-eradication’ periodic solution is globally attractive when harvesting for the prey is over certain value. Using a new qualitative analysis method for impulsive and delay differential equations, we prove the system is uniformly persistent when harvesting for the prey is under certain value. Further, we show the delay of digestion is a “profitless” time delay. Moreover, we show our theoretical results by numerical simulation. In this paper, the main feature is that we introduce a delay of digestion and impulsive effects into the predator-prey model and exhibit a new mathematical method which is applied to investigate the system which is governed by both impulsive and delay differential equations.     

Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

In this paper, we are concerned with a one‐dimensional porous‐thermoelastic system of type III with a viscoelastic damping and boundary time‐varying delay. Under suitable assumptions on relaxation function and time delay, we establish the exponential decay result of the system in which the damping is strong enough to stabilize the thermoelastic system in the presence of time delay. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks

This paper is concerned with the stability of n-dimensional stochastic differential delay systems with nonlinear impulsive effects. First, the equivalent relation between the solution of the n-dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding n-dimensional stochastic differential delay system without impulsive effects is established. Then, some stability criteria for the n-dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained. Finally, the stability criteria are applied to uncertain impulsive stochastic neural networks with time-varying delay. The results show that, this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks. Some examples are also discussed to illustrate the effectiveness of our theoretical results.     

A Comparison of Stochastic Systems with Different Types of Delays

In this article, we investigate the effects of different types of delays, a fixed delay and a random delay, on the dynamics of stochastic systems as well as their relationship with each other in the context of a just-in-time network model. The specific example on which we focus is a pork production network model. We numerically explore the corresponding deterministic approximations for the stochastic systems with these two different types of delays. Numerical results reveal that the agreement of stochastic systems with fixed and random delays depend on the population size and the variance of the random delay, even when the mean value of the random delay is chosen the same as the value of the fixed delay. When the variance of the random delay is sufficiently small, the histograms of state solutions to the stochastic system with a random delay are similar to those of the stochastic model with a fixed delay regardless of the population size. We also compared the stochastic system with a Gamma distributed random delay to the stochastic system constructed based on the Kurtz's limit theorem from a system of deterministic delay differential equations with a Gamma distributed delay. We found that with the same population size the histogram plots for the solution to the second system appear more dispersed than the corresponding ones obtained for the first case. In addition, we found that there is more agreement between the histograms of these two stochastic systems as the variance of the Gamma distributed random delay decreases.     

Properties of stationary states of delay equations with large delay and applications to laser dynamics

We consider properties of periodic solutions of the differential‐delay system, which models a laser with optical feedback. In particular, we describe a set of multipliers for these solutions in the limit of large delay. As a preliminary result, we obtain conditions for stability of an equilibrium of a generic differential‐delay system with fixed large delay τ. We also show a connection between characteristic roots of the equilibrium and multipliers of the mapping obtained via the formal limit τ→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

A uniform stability result for thermoelasticity of type III with boundary distributed delay

In this paper we consider a thermoelastic system of type III with boundary distributed delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Green and Naghdi's theory is still strong enough to uniformly stabilize the system even in the presence of time delay.     

Bifurcation analysis of a multidelayed HIV model in presence of immune response and understanding of in‐host viral dynamics

In this paper, we proposed a multidelayed in‐host HIV model to represent the interaction between human immunodeficiency virus and immune response. One delay was considered to incorporate the time required by the virus for various intracellular events to make a host cell productively infective, and the second delay was introduced to take into account the time required for adaptive immune system to respond against infection. We extensively analyzed this multidelayed model analytically and numerically. We show that delay may have both destabilizing and stabilizing effects even when the system contains a single immune response delay. It happens when there exists two sequences of critical values of this delay. If the system starts with stable state in absence of delay, then the smallest value of these critical delays causes instability and the next higher value causes stability. The system may also show stability switching for different values of the virus replication factor. These results demonstrate the possible reasons of intrapatients and interpatients variability of CD4⁺ T cells and virus counts in HIV‐infected patients.     

Asymptotic stability of positive equilibrium solution for a delayed prey–predator diffusion system

This article is concerned with a delayed Lotka–Volterra two-species prey–predator diffusion system with a single discrete delay and homogeneous Dirichlet boundary conditions. By applying the implicit function theorem, the asymptotic expressions of positive equilibrium solutions are obtained. And then, the asymptotic stability of positive equilibrium solutions is investigated by linearizing the system at the positive equilibrium solutions and analyzing the associated eigenvalue problem. It is demonstrated that the positive equilibrium solutions are asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than this critical value. In addition, it is also found that the system under consideration can undergo a Hopf bifurcation when the delay crosses through a sequence of critical values. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

Periodic Orbits for Some Systems of Delay Differential Equations

We provide sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay. We extend Kaplan-Yorke＇s method for finding periodic orbits from a delay differential equation with several delays to a system of delay differential equations with a unique delay.     

Delay-dependent and delay-independent stability criteria for a delay differential system

For a linear delay differential system with two coefficients and one delay, we establish some necessary and sufficient conditions on the asymptotic stability of the zero solution, which are composed of delay-dependent and delay-independent stability criteria. On the former criterion, the range of the delay is explicitly given.

     

Well-posedness and exponential stability of a flexible structure with second sound and time delay

In this paper, we consider a non-uniform flexible structure with time delay under Cattaneo's law of heat condition. We prove that the system is well-posed, and the system is exponential decay under a small condition on time delay.     

A delay differential equation model of HIV infection of CD4+ T-cells with cure rate

A delay differential equation as a mathematical model that described HIV infection of CD4⁺ T-cells is analyzed. When the constant death rate of infected but not yet virus-producing cells is equal to zero, the stability of the non-negative equilibria and the existence of Hopf bifurcation are investigated. A stability switch in the system due to variation of delay parameter has been observed, so is the phenomena of Hopf bifurcation and stable limit cycle. The estimation of the length of delay to preserve stability has been calculated. Further, when the constant death rate of infected but not yet virus-producing cells is not equal to zero, by using the geometric stability switch criterion in the delay differential system with delay dependent parameters, we present that stable equilibria become unstable as the time delay increases. Numerical simulations are carried out to explain the mathematical conclusions.     

Delay and partial system contents for a discrete-time G-D-c queue

We consider a discrete-time multiserver queueing system with infinite buffer size, constant service times of multiple slots and a first-come-first-served queueing discipline. A relationship between the probability distributions of the partial system contents and the packet delay is established. The relationship is general in the sense that it doesn’t require knowledge of the exact nature of the arrival process. By means of the relationship, results for the distribution of the partial system contents for a wide variety of discrete-time queueing models can be transformed into corresponding results for the delay distribution. As a result, a separate full analysis of the packet delay becomes unnecessary.      

Stability of linear systems with time-varying delays: A direct frequency domain approach

The stability of linear systems with uncertain bounded time-varying delays (without any constraints on the delay derivatives) is analyzed. It is assumed that the system is stable for some known constant values of the delays (but may be unstable for zero delay values). The existing (Lyapunov-based) stability methods are restricted to the case of a single non-zero constant delay value, and lead to complicated and restrictive results. In the present note for the first time a stability criterion is derived in the general multiple delay case without any constraints on the delay derivative. The simple sufficient stability condition is given in terms of the system matrices and the lengths of the delay segments. Different from the existing frequency domain methods which usually apply the small gain theorem, the suggested approach is based on the direct application of the Laplace transform to the transformed system and on the bounding technique in _L2$L_2$. A numerical example illustrates the efficiency of the method.