MEAN-SQUARE STABILITY OF NONLINEAR SYSTEMS WITH TIME-VARYING,RANDOM DELAY

A class of nonlinear systems with a time-varying delay is considered.The delay is modeled by a continuous-time Markov process with a finite number of states. Systems of this type may arise in real-time control applications. Employing a “delay-averaging” approach we demonstrate how certain mean-square stochastic stability conditions can be derived in terms of transition functions of the Markov process and stability properties of a system with a constant delay.     

Multiple‐interval‐dependent robust stability analysis for uncertain stochastic neural networks with mixed‐delays

This article deals with the problem of robust stochastic asymptotic stability for a class of uncertain stochastic neural networks with distributed delay and multiple time‐varying delays. It is noted that the reciprocally convex approach has been intensively used in stability analysis for time‐delay systems in the past few years. We will extend the approach from deterministic time‐delay systems to stochastic time‐delay systems. And based on the new technique dealing with matrix cross‐product and multiple‐interval‐dependent Lyapunov–Krasovskii functional, some novel delay‐dependent stability criteria with less conservatism and less decision variables for the addressed system are derived in terms of linear matrix inequalities. At last, several numerical examples are given to show the effectiveness of the results. © 2014 Wiley Periodicals, Inc. Complexity 21: 147–162, 2015     

HIV/AIDS Model with Delay and the Effects of Stochasticity

We present a deterministic HIV/AIDS model with delay. We then extend the model by adjoining terms capturing stochastic effects. The intensity of the fluctuations in the stochastic system is analytically evaluated using Fourier transform methods. We carry out simulations to assess differences in the dynamical behavior of the deterministic and stochastic models. Simulation results show that they are no significant differences in the behavior of the two models.     

Stochastic persistence and stability analysis of a modified Holling–Tanner model

The article aims to study the basic dynamical features of a modified Holling–Tanner prey–predator model with ratio‐dependent functional response. We have proved the global existence of the solution for the deterministic model. The parametric restriction for persistence of both species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided. The global dynamic behavior is examined thoroughly with supportive numerical simulation results. Next, we have formulated the stochastic model by perturbing the intrinsic growth rates of prey and predator populations with white noise terms. The existence uniqueness of solutions for stochastic model is established. Further, we have derived the parametric restrictions required for the persistence of the stochastic model. Finally, we have discussed the stochastic stability results in terms of the first and second order moments. Numerical simulation results are provided to support the analytical findings. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

网络化非周期采样控制系统的主动时间滞后控制: 随机脉冲切换系统方法

研究了一类带有随机丢包的非周期采样网络化控制系统的镇定问题.不同于传统观点往往将时滞看作系统稳定性的消极因素,考虑时间滞后对系统稳定性的积极影响, 并提出一个新颖的主动时间滞后控制方法来镇定该系统.为了分析时间滞后控制的积极作用并获得较低保守性的结论,首先把带随机丢包的非周期采样系统建模为带固定切换率的随机脉冲切换系统, 并在均方意义下提出一个新的分离引理用于分析随机脉冲切换系统的稳定性.然后,基于环 泛函方法和所提的分离引理,以线性矩阵不等式形式给出随机脉冲切换系统的均方稳定性判据.进一步,利用区间分割技术得到改进的均方稳定性判据.最后,利用一个经典的数值例子来验证所得稳定判据的有效性和所提方法的优势.     

Dynamical analysis of a delayed singular prey–predator economic model with stochastic fluctuations

This article studies a delayed singular prey–predator economic model with stochastic fluctuations, which is described by differential‐algebraic equations due to a economic theory. Local stability and Hopf bifurcation condition are described on the delayed singular prey–predator economic model within deterministic environment. It reveals the sensitivity of the model dynamics on gestation time delay. A phenomenon of Hopf bifurcation occurs as the gestation time delay increases through a certain threshold. Subsequently, a singular stochastic prey–predator economic model with time delay is obtained by introducing Gaussian white noise terms to the above deterministic model system. The fluctuation intensity of population and harvest effort are calculated by Fourier transforms method. Numerical simulations are carried out to substantiate these theory analysis. © 2013 Wiley Periodicals, Inc. Complexity 19: 23–29, 2014     

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.     

Impact of two types of time delays and cross-correlated multiplicative and additive noises on stability and stochastic resonance for a metapopulation system

In this paper, we investigate the stability and the shift between the extinction state and the stable one of a large density and the stochastic resonance (SR) for a metapopulation system subjected to two types of time delay terms, cross-correlation noises and multiplicative signal. By using the fast descent method and the method of small delay approximation, the expressions of the effective potential function and the signal-to-noise ratio (SNR) are obtained. We denote by Q the intensity of the multiplicative noise, and M the intensity of the additive noise, θ and τ the two time delay terms introduced into the metapopulation system. Our main results show some facts that time delay θ and the strength of correlation noise λ can restrain the development of the metapopulation, while the other term of time delay τ can accelerate the expansion of the population from the extinction state to the large stable one. We discover that it is possible to enhance the signal-to-noise ratio by adjusting the intensities of the multiplicative, additive noises and the time delays of the stochastic metapopulation system     

Stability of stochastic SIRS epidemic models with saturated incidence rates and delay

In this article, we consider stochastic susceptible-infected-removed-susceptible (SIRS) epidemic models with saturated incidence rates and delay. We investigate the stochastic stability in probability of the disease-free and endemic equilibria for the stochastic dynamic model with variability in the natural death rate, and the stochastic stability in probability of the endemic equilibrium for the dynamic model when the variability in the environment is proportional to a deviation between the state of the system and the endemic equilibrium. The numerical experiments are provided to support our theoretical results.     

Deterministic and stochastic analysis of a delayed allelopathic phytoplankton model within fluctuating environment

In this paper, we consider a two-dimensional model for two competitive phytoplankton species where one species is toxic phytoplankton and other is non-toxic species. The logistic growth of both the species follows the Hutchinson type growth law. First, we briefly discuss basic dynamical properties of non-delayed and delayed model system within deterministic environment. Next we formulate the stochastic delay differential equation model system to study the effect of environmental driving forces on the dynamical behavior. We calculate population fluctuation intensity (variance) for both species by Fourier transform method. Numerical simulations are carried out to substantiate the analytical findings. Significant results of our analytical findings and their interpretations from ecological point of view are provided in concluding section.     

Stability in terms of two measures of solutions to stochastic partial differential delay equations with switching

In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed‐index sequence monotonicity condition and under the average dwell‐time switching are considered.     

Exponential stability for markovian jumping stochastic BAM neural networks with mode‐dependent probabilistic time‐varying delays and impulse control

In this article, an exponential stability analysis of Markovian jumping stochastic bidirectional associative memory (BAM) neural networks with mode‐dependent probabilistic time‐varying delays and impulsive control is investigated. By establishment of a stochastic variable with Bernoulli distribution, the information of probabilistic time‐varying delay is considered and transformed into one with deterministic time‐varying delay and stochastic parameters. By fully taking the inherent characteristic of such kind of stochastic BAM neural networks into account, a novel Lyapunov‐Krasovskii functional is constructed with as many as possible positive definite matrices which depends on the system mode and a triple‐integral term is introduced for deriving the delay‐dependent stability conditions. Furthermore, mode‐dependent mean square exponential stability criteria are derived by constructing a new Lyapunov‐Krasovskii functional with modes in the integral terms and using some stochastic analysis techniques. The criteria are formulated in terms of a set of linear matrix inequalities, which can be checked efficiently by use of some standard numerical packages. Finally, numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 20: 39–65, 2015     

复合随机时滞系统的全局稳定

本文主要是研究了具有时滞随机复合系统的反馈律和全局稳定,及其所需要的充分条件.主要的方法是:引入一个测度函数u,使得关于ξ的随机系统稳定,再通过附加条件,从而达到整个复合系统的稳定.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

Temporal and spatiotemporal variations in a mathematical model of macrophage–tumor interaction

In the present paper we study a three-component mathematical model of tumor–immune system interaction. A number of solid tumors contain a high proportion of macrophages and these immune cells are known to have a remarkable impact on the progression and dormancy of such tumors. We assume these macrophages as the main immune system component facilitating tumor destruction. Stability criteria of the basic model around the steady state of coexistence are derived. Next, we consider the process of macrophage activation as non-instantaneous by using a distributed delay with a weak kernel and obtain a range for the macrophage death rate that ensures system stability. Finally, we incorporate the spatial irregularity of solid tumors by making the delay nonlocal. Analysis of the resulting spatiotemporal model gives a number of thresholds in terms of different system parameters that guarantee tumor stability. Numerical simulations are performed to justify analytical findings.     

Spin-Glass Stochastic Stability: a Rigorous Proof

We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spinglass quenched state. We show that stochastic stability holds in β-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V⁻¹. As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applied to the thermal fluctuations only. Communicated by Jennifer Chayes submitted 13/10/04, accepted 22/11/04     

Stochastic Epidemic SEIRS Models with a Constant Latency Period

In this paper, we consider the stability of a class of deterministic and stochastic SEIRS epidemic models with delay. Indeed, we assume that the transmission rate could be stochastic and the presence of a latency period of r consecutive days, where r is a fixed positive integer, in the “exposed” individuals class E. Studying the eigenvalues of the linearized system, we obtain conditions for the stability of the free disease equilibrium, in both the cases of the deterministic model with and without delay. In this latter case, we also get conditions for the stability of the coexistence equilibrium. In the stochastic case, we are able to derive a concentration result for the random fluctuations and then, using the Lyapunov method, to check that under suitable assumptions the free disease equilibrium is still stable.     

Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model

In this paper we consider the two species competitive delay plankton allelopathy stimulatory model system. We show the existence and uniqueness of the solution of the deterministic model. Moreover, we study the persistence of the model and the stability properties of its equilibrium points. We illustrate the theoretical results by some numerical simulations.