Blow-up behavior of Hammerstein-type delay Volterra integral equations

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations (DVIEs). Two types of delays, i.e., vanishing delay (pantograph delay) and non-vanishing delay (constant delay), are considered. With the same assumptions of Volterra integral equations (VIEs), in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. The blow-up behaviors of DVIEs with non-vanishing delay vary with different initial functions and the length of the lag, while DVIEs with pantograph delay own the same blow-up behavior of VIEs. Some examples and applications to delay differential equations illustrate this influence.     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

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.     

Modeling the role of constant and time varying recycling delay on an ecological food chain

We consider a mathematical model of nutrient-autotroph-herbivore interaction with nutrient recycling from both autotroph and herbivore. Local and global stability criteria of the model are studied in terms of system parameters. Next we incorporate the time required for recycling of nutrient from herbivore as a constant discrete time delay. The resulting DDE model is analyzed regarding stability and bifurcation aspects. Finally, we assume the recycling delay in the oscillatory form to model the daily variation in nutrient recycling and deduce the stability criteria of the variable delay model. A comparison of the variable delay model with the constant delay one is performed to unearth the biological relevance of oscillating delay in some real world ecological situations. Numerical simulations are done in support of analytical results.     

Global stability and Hopf bifurcation of a plankton model with time delay

A differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton–nutrient interaction. The differential inequalities’ methods and standard Hopf bifurcation analysis are applied. Some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state, respectively. It was shown that there is a Hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter.     

Control of Oversaturated Intersections

Simulation experiments are reported in which the control policies developed in an earlier paper for stable regimes are compared with other suggested procedures when employed during oversaturated periods. It is shown that the policies remain effective with respect to mean delay per vehicle. The problem of reducing the maximum individual delay may be tackled by means of a trade-off of mean delay by imposing a maximum phase duration during extreme congestion.     

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.     

Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay

To the best of the authors’ knowledge, there are no results based on the so-called Razumikhin technique via a general decay stability, for any type of stochastic differential equations. In the present paper, the Razumikhin approach is applied to the study of both pth moment and almost sure stability on a general decay for stochastic functional differential equations with infinite delay. The obtained results are extended to stochastic differential equations with infinite delay and distributed infinite delay. Some comments on how the considered approach could be extended to stochastic functional differential equations with finite delay are also given. An example is presented to illustrate the usefulness of the theory.     

Vibration suppression of a dynamical system to multi-parametric excitations via time-delay absorber

The aim of this work is to control the dynamic system behavior represented by a beam at simultaneous primary and sub-harmonic resonance condition, where the system damage is probable. Control is conducted via time delay absorber to suppress chaotic vibrations. A comprehensive investigation of the effect of the time delay on the control of a beam when subjected to multi- parametric excitation forces is presented. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation. Different resonance cases are reported and studied numerically. Stability of the steady state solution for the selected resonance case is investigated applying Rung-Kutta fourth order method and frequency response equations via Matlab 7.0 and Maple11. Time delay absorber is effective like ordinary one within a specified range of time delay. The delay time is an important factor in selecting the absorber. The effects of the different parameters of the absorber on the system behavior are studied numerically. The reported results are compared with the available published work.     

一类简化的时滞半导体激光方程的Hopf分岔

研究一类简化的时滞半导体激光方程的稳定性和Hopf分岔.以时滞量为参数,分析系统线性化方程零解的稳定性,给出系统产生Hopf分岔临界时滞表达式,最后用数值模拟对结论进行验证.     

Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay

In this paper, we investigate the permanence of an SIR epidemic model with a density-dependent birth rate and a distributed time delay. We first consider the attractivity of the disease-free equilibrium and then show that for any time delay, the delayed SIR epidemic model is permanent if and only if an endemic equilibrium exists. Numerical examples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay.     

The effect of time delay on the dynamics of an SEIR model with nonlinear incidence

In this paper, a SEIR epidemic model with nonlinear incidence rate and time delay is investigated in three cases. The local stability of an endemic equilibrium and a disease-free equilibrium are discussed using stability theory of delay differential equations. The conditions that guarantee the asymptotic stability of corresponding steady-states are investigated. The results show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation when using the time delay as a bifurcation parameter. Applying the normal form theory and center manifold argument, the explicit formulas determining the properties of the bifurcating periodic solution are derived. In addition, the effect of the inhibitory effect on the properties of the bifurcating periodic solutions is studied. Numerical simulations are provided in order to illustrate the theoretical results and to gain further insight into the behaviors of delayed systems.     

Dynamical behavior of a delay virus dynamics model with CTL immune response

In this paper, the dynamical behavior of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation around immune-free equilibrium and endemic equilibrium to occur are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

Bifurcation analysis of delay-induced periodic oscillations

In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude and a constant frequency; the number of solutions increases with the size of the delay. Indeed, for many physical applications in which oscillatory instabilities are induced by a delayed response or feedback mechanism, the system under consideration forms the underlying backbone for a mathematical model. Our study showcases the effectiveness of performing a numerical bifurcation analysis, alongside the use of analytical and geometrical arguments, in investigating systems with delay. We identify curves of codimension-one bifurcations of periodic solutions. We show how these curves interact via codimension-two bifurcation points: double singularities which organise the bifurcations and dynamics in their local vicinity.     

非线性时滞差分方程的持续生存和渐近性质

研究了一类非线性时滞差分方程解的渐近性质,得到了方程持续生存和全局吸引的充分条件.这些结果可应用于一类非线性时滞差分方程和时滞离散Logistic模型,并包含了一些已知的结果.     

Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary

In this work, we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary. We construct a reaction–diffusion problem with delay in the interior, where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter ? goes to zero. We analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary. This convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary.     

Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays

In this paper, the problem of exponential stabilization for a class of linear systems with time-varying delay is studied. The time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is not necessary to be differentiable. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the exponential stabilization of the systems are first established in terms of LMIs. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.     

Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling

In this paper we propose a reaction-diffusion system with twodistributed delays to stimulate the growth of plankton communitiesin the lakes/oceans in which the plankton feeds on a limitingnutrient supplied at a constant rate. The limiting nutrientis partially recycled after the death of the organisms and adistributed delay is used to model nutrient recycling. The seconddelay is involved in the growth response of the plankton tonutrient uptake. We first show that there are oscillations (Hopfbifurcations) in the delay model induced by the second delay.Then we study Turing (diffusion-driven) instability of the reaction-diffusionsystem with delay. Finally, it is shown that if the delay modelhas a stable periodic solution, then the corresponding reaction-diffusionmodel with delay has a family of travelling waves.     

Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback

The resonance and chaos of micro (nano) electro mechanical resonators with time delay feedback is concerned in the paper. Based on the experimental results, a lumped single degree-of-freedom (1DOF) model is studied and the effects of time delay displacement and velocity feedback on the system are investigated. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The Melnikov function method is then extended to the two time delay systems, and the analytically required condition for chaos was obtained. Finally, the fourth-order Runge–Kutta method, point-mapping method and spectrum diagram are used to simulate the evolution of the dynamic behavior of the time delay control system. Also, the stability of this time delay control system is studied thoroughly. The results show that time delay feedback is a good method for the control system and that reasonable selection of control system parameters can effectively suppress the vibration level for micro/nano-electro-mechanical resonator systems.     

Asymptotic behaviour of the solutions of a nonautonomous linear delay difference system

It is proved under appropriate assumptions that the solutions of a linear system of nonautonomous delay difference equations have finite limit at infinity. The results are based on a transformation of the delay difference system into a first-step recursion, where the companion matrices are well treatable from our point of view. Our theory is illustrated by examples, including a class of linear delay difference equations with unbounded coefficients.