Alexander V. Rezounenko
A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss advantages of this class. Local and long-time asymptotic properties, including the existence of global attractor, are studied.
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Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue-Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.
A wide class of non-autonomous nonlinear parabolic partial differential equations with delay is studied. We allow in our investigations different types of delays such as constant, time-dependent, state-dependent (both discrete and distributed) to be presented simultaneously. The main difficulties arise due to the presence of discrete state-dependent delays since the nonlinear delay term is not Lipschitz on the space of continuous functions. We find conditions for the local existence, uniqueness and study the invariance principle.
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of \textit{the solution manifold} proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C\sp 1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, {195}(1), (…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.
This work is the first attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.
This paper is concerned with the traveling wave solutions of delayed reaction-diffusion systems. By using Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka-Volterra competition systems with distributed delays are established…
We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Such models frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent attracting set and also demonstrate that the equation with a distributed …
We consider a linear scalar delay differential equation (DDE), consisting of two arbitrary distributed time delays. We formulate necessary conditions for stability of the trivial solution which are independent of the distributions. For the case of one discrete and one gamma distributed delay, we give an explicit description of the region of stability of the trivial solution and discuss how this depends on the model parameters.
Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillation around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that the linear equati…
Period two solution for a class of distributed delay differential equations
本文研究一类分布时滞微分方程的二周期解存在性，证明该解可由哈密顿系统给出，并基于Kaplan-Yorke方法，以Jacobi椭圆函数实例说明。
We consider a system of several nonlinear equations with a distributed delay and obtain absolute asymptotic stability conditions, independent of the delay. The ideas of the proofs are based on the notion of a strong attractor. The results are applied to Hopfield neural networks, Nicholson's blowflies type system, and compartment models of population dynamics.
Distributed delay equations have been used to model situations in which there is some sort of delay whose duration is uncertain. However, the interpretation of a distributed delay equation is actually very different from that of a delay differential equation with a random delay. This work explicitly highlights this distinction as it is an important consideration to make when modeling delayed systems in which the delay can take on several values.
Heterogeneous delays with positive lower bound (gap) are taken into consideration in Kuramoto oscillators. We first establish a perturbation technique, by which universal normal forms and detailed dynamical behavior of this model can be obtained easily. Theoretically, a hysteresis loop is found near the subcritically bifurcated coherent state on the Ott-Antonsen's manifold. For Gamma distributed delay with fixed variance and mean, we find large gap destroys the loop and significantly increases in the number of coexisted coherent attractors. This result is also explained in the viewpoint of exc…
This contribution presents two exponential stability criteria for linear systems with multiple pointwise and distributed delays. These results (necessary and sufficient conditions) are given in terms of the delay Lyapunov matrix and the fundamental matrix of the system. An important property is that the stability test requires a finite number of mathematical operations.
Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean…
Distributed Delay Systems with Oscillations in the Time Histories
针对时间历程含振荡的分布时滞动力系统，提出基于 Linear Chain Trick（LCT）的算法，并以种群动力学示例说明其应用。
The objective of this paper is to investigate the stability of limit cycles of a mathematical model with a distributed delay which describes the interaction between p53 and mdm2. Choosing the delay as a bifurcation parameter we study the direction and stability of the bifurcating periodic solutions using the normal form and the center manifold theorem. Some numerical examples are finally made in order to confirm the theoretical results.
Spectral comparison of compound cocycles generated by delay equations in Hilbert spaces
本文提出了一种在希尔伯特空间中比较由延迟方程生成的复合协同体谱性质的方法，用于分析非线性问题中的全局稳定性。
$L^q$ approximate controllability frequency criterion for linear difference delay equations with distributed delays
该研究基于代数视角与Y. Yamamoto的实现理论，提出带分布时滞的线性差分时滞方程有限时间$L^q$近似能控性的频域充要判据，并得到其最小能控时间的上界。
State dependent delay differential equations with distributed memory
本文研究了具有分布式记忆的状态依赖延迟微分方程，通过分析逻辑模型展示了高阶记忆核可能引发振荡动力学，并建立了稳定性与分支分析的严格框架。
A distributed-delay model for the El Niño Southern Oscillation with a minimum delay: a case study of the shifted linear chain trick
本研究提出移位厄兰核与线性链技巧，对具有最小延迟的厄尔尼诺-南方涛动分布式延迟模型进行分岔分析，揭示不同分布宽度下的共振结构。
In this paper, we consider a continuous-time model with discrete and dis-tributed delays to describe how two pieces of information interact in online social networks. Sufficient conditions are carried out to illustrate the stability of each e-quilibrium. Taking time delay as a bifurcation parameter, the system undergoes a sequence of Hopf bifurcation when this parameter passes through a critical value. By methods of multiple scales, we prove that the direction of Hopf bifurcation is depending on the condition which is related to delay.
Sparse identification of delay equations with distributed memory
本文提出了一种用于具有分布式延迟和再生素方程的延迟微分方程的稀疏识别方法，通过稀疏回归直接重建核函数，提升数据驱动发现的准确性与可解释性。
In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous boundary Neumann condition, which have a positive steady state. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument of a sub and super-solution combined with fluctuation method. We also give a condition for which the exponential stability of the positive steady state is reached. As an example, we apply our results to diffusive Nicholson blowflies and diffusive Mackey-Glass equation with distributed delay. We …
Macroscopic Multistability and Bifurcations in Theta-Neuron Networks with Distributed Delays
研究具有分布式延迟的θ神经元网络，通过渡边-斯特罗加茨约化等方法推导出复序参量的延迟微分方程，得到平衡点稳定性准则和霍普夫分岔条件，数值模拟验证结果，揭示延迟对网络稳定性的多种影响。
Influence of Large Mean Delay on Distributed Delay Differential Equations Dynamics: Application to a Neural Mass Model
本文研究大平均时延下分布式时滞微分方程的谱结构，提出谱分裂为强临界谱、渐近强谱和伪连续谱，并应用于Wilson-Cowan模型的稳定性与分岔分析。
Asymptotic stability and exponential stability for a class of impulsive neutral differential equations with discrete and distributed delays
本文基于不动点理论，不依赖 Lyapunov 函数，研究了一类具离散和分布时滞的脉冲中立型微分方程的渐近与指数稳定性，给出充分条件并改进已有结果。
In the presented paper a family of angiogenesis models, that is a generalisation of Hahnfeldt et al. model is proposed. Considered family of models is a system of two differential equations with distributed time delays. The global existence and the uniqueness of the solutions are proved. Moreover, the stability of the unique positive steady state is examined in the case when delay distributions are Erlang or piecewise linear distributions. Theorems guaranteeing the existence of stability switches and occurrence of the Hopf bifurcation are proved. Theoretical results are illustrated by numerica…
We derive sufficient conditions for exponential decay of solutions of the delay negative feedback equation with distributed delay. The conditions are written in terms of exponential moments of the distribution. Our method only uses elementary tools of calculus and is robust towards possible extensions to more complex settings, in particular, systems of delay differential equations. We illustrate the applicability of the method to particular distributions - Dirac delta, Gamma distribution, uniform and truncated normal distributions.
We study a variant of the Cucker-Smale system with distributed reaction delays. Using backward-forward and stability estimates on the quadratic velocity fluctuations we derive sufficient conditions for asymptotic flocking of the solutions. The conditions are formulated in terms of moments of the delay distribution and they guarantee exponential decay of velocity fluctuations towards zero for large times. We demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking condition can be resolved ana…
Stability of coupled Wilson-Cowan systems with distributed delays
本文研究两个耦合 Wilson-Cowan 对的稳定性，比较离散时滞与弱 Gamma 分布时滞对病理振荡起止机制的影响，并通过基底神经节和前额叶-边缘系统模拟验证分析结果。
Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory
本文结合 Galerkin 方法与 Dafermos 变换研究带长时间记忆的加性噪声随机非局部 PDE，证明解的适定性与随机吸引子存在性，并建立 Wong-Zakai 逼近的收敛及吸引子上半连续性。
This paper studies the effects of coupling with distributed delay on the suppression of oscillations in a system of coupled Stuart-Landau oscillators. Conditions for amplitude death are obtained in terms of strength and phase of the coupling, as well as the mean time delay and the width of the delay distribution for uniform and gamma distributions. Analytical results are confirmed by numerical computation of the eigenvalues of the corresponding characteristic equations. These results indicate that larger widths of delay distribution increase the regions of amplitude death in the parameter spac…
Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with non-linear transit rates and possibly delayed arguments are also equivalent to a scalar distributed delay differential equation. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute…
Ordinary differential equations (ODE) models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models…
Mapping dynamical systems with distributed time delays to sets of ordinary differential equations
本文提出 kernel series framework，将离散指数 gamma 型分布时滞精确映射为 $N+1$ 个常微分方程，并揭示分布延迟对 Mackey-Glass 分岔、混沌及混沌扩散的不同影响。
Classical distributed estimation scenarios typically assume timely and reliable exchanges of information over the sensor network. This paper, in contrast, considers single time-scale distributed estimation via a sensor network subject to transmission time-delays. The proposed discrete-time networked estimator consists of two steps: (i) consensus on (delayed) a-priori estimates, and (ii) measurement update. The sensors only share their a-priori estimates with their out-neighbors over (possibly) time-delayed transmission links. The delays are assumed to be fixed over time, heterogeneous, and kno…
In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form the…
Time-Delay Systems with Discrete and Distributed delays: Discontinuous Initial Conditions and Reachability Sets
本文研究了包含混合点wise和分布延迟的时间滞后系统，提出新的充分条件以保证可达集有界，并识别了满足这些条件的广泛系统类。
We study the global dynamics of a reaction-diffusion SEIR infection model with distributed delay and nonlinear incidence rate. The well-posedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
Analyzing Smoothness and Dynamics in an SEIR$^{\text{T}}$R$^{\text{P}}$D Endemic Model with Distributed Delays
本文研究了含分布时滞的SEIRⁿᵉᵐᵖᵒʳᵃʳʸ Rᵖᵉʳᵐᵃⁿᵉⁿᵗ Dᵉᵃᵗʰ地方性模型，引入紧支集连续积分核时滞，证明解的光滑性、有界性和非负性，数值模拟验证其可由离散模型近似，助力传染病动力学理解与数值求解。
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are not currently available. Accordingly, modellers often resort to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations. In this work, we develop a functionally continuous Runge-Kutta method to numerically integrate the gamma distributed DDE and perform numerical tests to confirm the accuracy of the numeric…
We propose a distributed (single) target tracking scheme based on networked estimation and consensus algorithms over static sensor networks. The tracking part is based on linear time-difference-of-arrival (TDOA) measurement proposed in our previous works. This paper, in particular, develops delay-tolerant distributed filtering solutions over sparse data-transmission networks. We assume general arbitrary heterogeneous delays at different links. This may occur in many realistic large-scale applications where the data-sharing between different nodes is subject to latency due to communication-reso…
The paper addresses the problem of calculating the noise-induced switching rates in systems with delay-distributed kernels and Gaussian noise. A general variational formulation for the switching rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions represent the most probable, or optimal, path, which maximizes the probability of escape. Explicit analytical results for the switching rates for small mean time delays are obtained for the uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution …
Stabilizing the Staking Rate, Dynamically Distributed Inflation and Delay Induced Oscillations
本文研究了动态分配通胀机制中因反馈延迟引发的质押率振荡问题，提出了一种新型分配模型，有效抑制了振荡并稳定了目标质押范围内的收益率。
Stability and Convergence of Distributed Stochastic Approximations with large Unbounded Stochastic Information Delays
本文将Borkar-Meyn稳定性定理推广到具有任意矩界信息延迟的分布式随机逼近，通过引入信息年龄过程并利用新的Gronwall型不等式证明稳定性，应用于分布式优化和动量SAs分析。
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