Leonid Berezansky, Elena Braverman
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.
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Global attractivity criteria for a discrete-time Hopfield neural network model with unbounded delays via singular M-matrices
针对含无限延迟和泄漏项延迟的离散时间非自治 Hopfield 神经网络，本文利用不必可逆及不可约奇异 M-矩阵分别建立有界与无界激活函数下的全局吸引性准则，并通过数值模拟验证其有效性。
Global Attractivity of a Nonlocal Delayed Diffusive Dengue Model in a Spatially Homogeneous Environment
本文通过李雅普诺夫函数证明了非局部时滞扩散登革热模型在空间均质环境中的正稳态全局吸引性，无需额外条件，且 Re_0 大于1时成立。
Using properties of asymptotically almost periodic solutions we prove existence theorem for piece-wise continuous almost periodic solutions of differential equations with delay and impulses. We apply these results to study almost periodic single species model with stage structure and impulses.
In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original syste…
The paper deals with a multiple species Lotka-Volterra model with infinite distributed delays and feedback controls, for which we assume a weak form of diagonal dominance of the instantaneous negative intra-specific terms over the infinite delay effect in both the population variables and controls. General sufficient conditions for the existence and attractivity of a saturated equilibrium are established. When the saturated equilibrium is on the boundary of $\R^n_+$, sharper criteria for the extinction of all or part of the populations are given. While the literature usually treats the case of…
The problem considered in the paper is exponential stability of linear equations and global attractivity of nonlinear non-autonomous equations which include a non-delay term and one or more delayed terms. First, we demonstrate that introducing a non-delay term with a non-negative coefficient can destroy stability of the delay equation. Next, sufficient exponential stability conditions for linear equations with concentrated or distributed delays and global attractivity conditions for nonlinear equations are obtained. The nonlinear results are applied tothe Mackey-Glass model of respiratory dyna…
In this paper, we study the global attractivity for a class of periodic difference equation with delay which has a generalized form of Pielou's difference equation. The global dynamics of the equation is characterized by using a relation between the upper and lower limit of the solution. There are two possible global dynamics: zero solution is globally attractive or there exists a periodic solution which is globally attractive. Recent results in [E. Camouzis, G. Ladas, Periodically forced Pielou's equation, J. Math. Anal. Appl. 333 (1) (2007) 117-127] is generalized. Two examples are given to …
A subset $A$ of the state space is called uniformly globally weakly attractive if for any neighborhood $S$ of $A$ and any bounded subset $B$ there is a uniform finite time $τ$ so that any trajectory starting in $B$ intersects $S$ within the time not larger than $τ$. We show that practical uniform global asymptotic stability (pUGAS) is equivalent to the existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in…
In this paper, a non-autonomous stochastic logistic system is considered. An interesting result on the effect of stochastically perturbation for the dynamic behavior are obtained. That is, under certain conditions the stochastic system have similar dynamic behave with the deterministic system, but if the noise is sufficiently large, it will spoil these nice properties. Furthermore, we introduce a new research method for studying the stochastic equation and some new sufficient conditions for the stochastic bounded, stochastic permanence, extinction and global attractivity of the system are esta…
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 …
A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is shown that typical behaviour in the system…
The present paper deals with autonomous integral equations with infinite delay via dynamical system approach. Existence, local exponential attractivity, and other properties of center manifold are established by means of the variation-of-constants formula in the phase space that is obtained in a previous paper \cite{mur}. Furthermore, we prove a stability reduction principle by which the stability of an autonomous integral equation is implied by that of an ordinary differential equation which we call the "central equation".
We study some aspects of the global dynamics of an $n$-dimensional Lotka-Volterra system with infinite delay and patch structure, such as extinction, persistence, existence and global attractivity of a positive equilibrium. Both the cases of an irreducible and reducible linear community matrix are considered, and no restriction on the signs of the intra- and inter-specific delayed terms is imposed. Although the system is not cooperative, our approach often uses comparison results applied to an auxiliary cooperative system. Some models in recent literature are generalised, and results improved.
In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate t…
The delayed logistic equation (also known as Hutchinson's equation or Wright's equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes te…
Bifurcation of periodic and antiperiodic solutions in non-autonomous potential-type delay systems
研究五类多延迟非自治势时滞微分系统周期解和反周期解的分岔，通过将系统重述为广义哈密顿系统，应用哈密顿分岔理论建立了法德尔 - 拉宾诺维茨型和拉宾诺维茨型的分岔结果。
Integral representation formula for linear non-autonomous difference-delay equations
本文证明了一类线性非自治差分时滞系统连续解的常数变易型积分表示公式，利用基本解与初始数据的Lebesgue-Stieltjes积分给出精确形式，并将其推广到时变情形，为时滞系统稳定性研究提供基础。
Bifurcations of periodic and antiperiodic orbits near an equilibrium in autonomous differential delay systems with one or two delays
研究四类含一或两个时滞的自治微分时滞方程平衡点附近周期和反对称解的分支，通过转化为哈密顿系统，利用哈密顿分支理论及马斯洛夫型指标计算，得到相关分支结果。
We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide resu…
Existence and spatio-temporal symmetric patterns of periodic solutions to second order reversible equivariant non-autonomous periodic systems with multiple delays are studied under the Hartman-Nagumo growth conditions. The method is based on using the Brouwer $D_1 \times \mathbb Z_2\times Γ$-equivariant degree theory, where $D_1$ is related to the reversing symmetry, $\mathbb Z_2$ is related to the oddness of the right-hand-side and $Γ$ reflects the symmetric character of the coupling in the corresponding network. Abstract results are supported by a concrete example with $Γ= D_n$ -- the dihedr…
The exponential ordering is exploited in the context of non-auto\-no\-mous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost perio…
Stability analysis of Richardson models with delay for confrontation between two countries
本文提出含延迟的非自治数学模型，研究两国对抗平衡态的稳定性，分析宿敌因素对稳定性的影响，并探讨自治与非自治模型的渐近稳定性及Hopf分岔现象。
In this paper, a class of Kolmogorov systems with delays are studied. Sufficient conditions are provided for a system to have a compact uniform attractor. Then Jansen's result (J. Math. Biol. Vol. 25 (1987) 411-422) for autonomous replicator and Lotka-Volterra systems has been extended to delayed nonautonomous Kolmogorov systems with periodic or autonomous Lotka-Volterra subsystems. Thus, simple algebraic conditions are obtained for partial permanence and permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays.
Shadowing, Hyers--Ulam stability and hyperbolicity for nonautonomous linear delay differential equations
本文证明了对具有一致有界系数的非自治线性延迟微分方程，双曲性、Hyers--Ulam稳定性和阴影性三者等价，并通过例子说明有界性假设的必要性。
Pullback measure attractors and limiting behaviors of McKean-Vlasov stochastic delay lattice systems
本文研究了整数集上非自治 McKean-Vlasov 随机时滞格点系统解分布的长期行为，证明了解的适定性与拉回测度吸引子的存在唯一性，并揭示了不变测度的遍历性、指数混合性及吸引子的上半连续性。
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.
We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.
Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay
本文证明了具有指数二分性的紧致概自守矩阵系统的关联系统同样具有指数二分性，并揭示了其 Green 函数的紧致双概自守性，进而研究了带混合时滞的生物非自治微分方程正紧致概自守解的存在性。
The aim of this paper is to study the dynamical behavior of non-autonomous stochastic hybrid systems with delays. By general Krylov-Bogolyubov's method, we first obtain the sufficient conditions for the existence of an evolution system of measures of the non-autonomous stochastic system and also give some easily verifiable conditions. We then prove a sufficient condition for convergence of evolution systems of measures as the delay approaches zero. As an application of the abstract theory, we first prove the existence of evolution systems of measures for stochastic system with time-vary delays…
A non-autonomous evolution semi-linear differential system under non-instantaneous impulses, delays, and perturbed by non-local conditions is studied. Its piece-wise continuous solutions belong to a finite-dimensional Banach space. The existence and uniqueness of solutions on the interval $[-r,τ]$ are obtained by applying Karakostas fixed-point theorem. Further results concerning solution prolongation are developed. An example is presented, and several remarks on the infinite-dimensional case are included.
In this paper, we study the asymptotic behavior of solution to a non-autonomous diffusion equations with delay containing some hereditary characteristics and nonlocal diffusion in time-dependent space $C_{\mathcal{H}_{t}(Ω)}$. When the nonlinear function $f$ satisfies the polynomial growth of arbitrary order $p-1$ $(p \ge 2)$ and the external force $h \in L_{l o c}^{2}\left(\mathbb{R} ; H^{-1}(Ω)\right)$, we establish the existence and regularity of the time-dependent pullback attractors.
Pullback Attractors for a Non-Autonomous Plate System with Strong Damping and Delay
本文针对含时滞的非线性非自治强阻尼板方程，通过构造修正能量泛函和压缩函数法，证明了其拉回吸引子的存在性。
Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times Γ\times \mathbb Z_2$-equivariant degree theory, where $O(2)$ is related to the reversing symmetry, $Γ$ reflects the symmetric character of the coupling in the corresponding network and $\mathbb Z_2$ is related to the oddness of the right-hand-side. Abstract results are supported by a concrete example with $Γ= D_6$ -- the dihedral group of order 12.
An exact solution for a non-autonomous delay differential equation
本文提出了一种非自治延迟微分方程的精确解法，通过高斯形状动力学的和形式，首次给出了此类方程的显式解，并揭示了系统动力学的包络特性。
This paper provides a dynamical frame to study non-autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type II over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.
For equations with the neutral term strictly delayed we construct the fundamental solution, derive a variation-of-constants formula for inhomogeneous equations, and prove growth estimates. Only unavoidable measure and integration theory, up to the Riesz Representation Theorem, is used. A key notion is pointwise convergence of bounded sequences of continuous functions.
Convergence of asymptotic systems in Cohen-Grossberg neural network models with unbounded delays
本文研究含无限离散时变时滞与分布时滞的非自治Cohen-Grossberg神经网络模型，在非时滞项渐近支配时滞项的条件下证明渐近系统收敛性，并通过例子和数值模拟验证结果。
Existence and regularity of pullback attractors for nonclassical non-autonomous diffusion equations with delay
本文研究了临界增长条件下具时滞非自治扩散方程弱解在时变空间中的渐近行为，利用 Faedo-Galerkin 近似法证明解的适定性，并通过能量估计建立了拉回吸引子的存在性与正则性。
Delay-induced dynamics in a nonlinear crime interaction model with periodic forcing
提出一个非线性时滞模型描述犯罪与非犯罪人群的交互动力学，结合社会影响机制、Holling II型功能反应饱和效应和时变执法行动，分析延迟诱导的稳定性切换和周期解存在性。
For a wide class of second order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal, allows to stabilize unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilization by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.
For autonomous delay differential equations $x'(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$, $t\ge0$, on open subsets of the Fréchet space $C((-\infty,0],\mathbb{R}^n)$. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application we obtain processes which incorporate all solutions of Volterra integro-differential equations $x'(t)=\int_0^tk(t,s)h(x(s))ds$
Generalized non-autonomous Cohen-Grossberg neural network model
本文研究具有无界分布时滞的广义高阶周期Cohen-Grossberg神经网络模型，利用非时滞项主导性条件和重合度定理，建立了周期解存在性与全局指数稳定性的判据，并通过数值模拟验证其有效性。
Exponential dichotomy for a class of asymptotically autonomous delayed differential equations
本文为一类渐近自治时滞微分方程建立了在整个实轴上存在指数二分性的判据，核心方法是借助加权 Sobolev 空间使相关微分算子成为 Fredholm 算子并利用其性质完成证明。
For the nonlinear second order Lienard-type equations with time-varying delays $$ \ddot{x}(t)+\sum_{k=1}^m f_k(t,x(t),\dot{x}(g_k(t)))+\sum_{k=1}^l s_k(t,x(h_k(t)))=0, $$ global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for relevant linear equations and are applied to derive explicit stability conditions for the nonlinear Kaldor-Kalecki business cycle model. We also explore multistability of the sunflower non-autonomous equation and its modifications.
Periodic Solutions of a Delay Differential Equation with a Periodic Multiplier
本文研究带周期乘子的时滞非自治标量微分方程，证明存在两类稳定慢振荡周期解，分段常数情形下可显式构造，且小光滑扰动下周期动力学可保持，结果经数值模拟验证。
Periodic patterns in simple biological differential delay models
针对带周期乘法系数和延迟负反馈的标量微分延迟方程，本文将动力学归约为区间映射，证明两类慢振荡周期解存在，并用数值计算验证。
We consider a nonlinear non-autonomous system with time-varying delays $$ \dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t))) $$ which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed by the system's parameters, is an $M$-matrix. Comparison with the tests obtained by K. Gopalsamy (2007) and B. Liu (2013) fo…
This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a function of the state variable. A delay map with constant delays can be rewritten explicitly as a discrete time dynamical system on an appropriate function space, and a delay map with small state dependent terms can be viewed as a ``non-autonomous''…
We show the existence of nonautonomous invariant manifolds for planar, asymptotically autonomous differential equations, that have equilibrium solutions with zero Lyapunov spectrum. These invariant manifolds correspond to the stable and unstable manifold of a desingularized equation, that we obtain by using the blow-up method. More precisely, the blow-up method is extended to the nonautonomous setting and transforms the original finite-dimensional ordinary differential equation into an infinite-dimensional delay equation with infinite delay, but hyperbolic structure. In the technical construct…
Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates
本文在希尔伯特空间中研究时滞方程，构造非线性半群与求解算子，方法更初等，并探讨维数估计与惯性流形。
This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics. The proofs are based on a Perron type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtaine…
Data-Driven Reduced Modeling of Delayed Dynamical Systems via Spectral Submanifolds
本文提出了一种基于谱子流形理论的数据驱动方法，用于减少非线性延迟系统的模型复杂度，通过比较基于方程和数据驱动的SSM降维方法，展示了该方法在处理延迟微分方程时的有效性，并应用于多个非线性延迟系统，得到无延迟的ODE模型。
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