Asymptotic properties of solutions to linear nonautonomous delay differential equations through generalized characteristic equations

Smoothness of Topological Equivalence on the Half Line for Nonautonomous Systems

We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a uniformly continuous homeomorphism inspired in the Palmer's one restricted to the positive half line, providing sufficient conditions ensuring its $C^{r}$--smoothness. Additionally, we study the preservation of the uniform stability properties by this homeomorphism.

Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line

A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the positive half--line while the nonlinearity has some boundedness and Lipschitzness conditions. As a consequence, we study the asymptotical stability and its preservation by topological equivalence.

Periodic perturbations with delay of separated variables differential equations

We study the structure of the set of harmonic solutions to perturbed nonautonomous, T-periodic, separated variables ODEs on manifolds. The perturbing term is allowed to contain a finite delay and to be T-periodic in time.

Asymptotic behavior of coupled dynamical systems with multiscale aspects

We study the asymptotic behavior, as time t goes to infinity, of nonautonomous dynamical systems involving multiscale features. These systems model the emergence of various collective behaviors in game theory, as well as the asymptotic control of coupled sytems.

Existence and multiplicity of periodic solutions for a hematopoiesis model

A general nonautonomous Mackey-Glass equation for the regulation of the hematopoiesis with several non-constant delays is studied. Using topological degree methods, we prove the existence and multiplicity of positive periodic solutions.

New Stability Conditions for Linear Differential Equations with Several Delays

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m a_k(t)x(h_k(t))=0 $$ with measurable delays and coefficients. These results are compared to known stability tests.

Existence and Stability of Positive Periodic Solutions to Functional Differential Equations

We establish several delay-independent criteria for the existence and stability of positive periodic solutions of n-dimensional nonautonomous functional differential equation by several fixed point theorems. Examples from positive and negative growth feedback systems demonstrate the results.

延迟方程中的可容许性刻画及指数二分法对小延迟扰动的鲁棒性

Admissible characterization in delay equations and robustness of exponential dichotomies against small-delay perturbations

本文针对非自治延迟方程，提出基于两对Banach空间的可容许性刻画，得到显式二分指数，并借助算子扰动方法证明小延迟扰动保持指数二分法。

Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear ra…

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form \label{semilineq} \frac{dx}{dt}= A(t)x+f(t,x). Here $A(t)$ is a (possibly unbounded) linear operator acting on a real or complex Banach space $\X$ and $f: \R\times\X\to\X$ is a (possibly nonlinear) continuous function. We assume that the linear equation \eqref{lineq} is well-posed (i.e. there exists a continuous linear evolution family \Uts such that for every $s\in\R_+$ and $x\in D(A(s))$, the function $x(t) = U(t, s) x$ is the uniquely determined solution of equation \eqref{lineq} satisfying $x(s) = x$). Then we can conside…

New criterion of asymptotic stability for delay systems with time-varying structures and delays

In this paper, we study asymptotic stability of the zero solution of a class of differential systems governed by a scalar differential inequality with time-varying structures and delays. We establish a new generalized Halanay inequality for the asymptotic stability of the zero solution for such systems under more relaxed conditions than the existing ones. We also apply the theoretical results to the analysis of self synchronization in networks of delayed differential systems and obtained a more general sufficient condition for self synchronization.

Periodic and Asymptotically Periodic Solutions of Systems of Nonlinear Difference Equations with Infinite Delay

In this work we study the existence of periodic and asymptotically periodic solutions of a system of nonlinear Volterra difference equations with infinite delay. By means of fixed point theory, we furnish conditions that guarantee the existence of such periodic solutions.

无穷远处动力学中具有法向双曲性质的 ODE 的爆破行为

Blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity

本文通过无穷远处动力学描述 ODE 的爆破行为，利用相空间闭嵌入和时间尺度变换将爆破描述转化为地平线上紧轨迹的跟踪性质，并证明具有渐近相位的双曲不变流形可诱导爆破。

Dichotomies and asymptotic equivalence in alternately advanced and delayed differential systems

In this paper, ordinary and exponential dichotomies are defined in differential equations with equations with piecewise constant argument of general type. We prove the asymptotic equivalence between the bounded solutions of a linear system and a perturbed system with integrable and bounded perturbations.

Stability of scalar nonlinear fractional differential equations with linearly dominated delay

In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.

非振荡区域中带反应型时滞的Cucker-Smale模型的渐近蜂拥

Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime

该研究针对带反应型时滞的Cucker-Smale系统，通过后向-前向与稳定性估计推导渐近蜂拥的充分条件，在双智能体简化场景下得到非振荡时滞的精确条件，并评述宏观描述的数学问题。

Growth Rates of Solutions of Superlinear Ordinary Differential Equations

In this letter we obtain sharp estimates on the growth rate of solutions to a nonlinear ODE with a nonautonomous forcing term. The equation is superlinear in the state variable and hence solutions exhibit rapid growth and finite-time blow-up. The importance of ODEs of the type considered here stems from the key role they play in understanding the asymptotic behaviour of more complex systems involving delay and randomness.

Asymptotic behaviour for a class of non-monotone delay differential systems with applications

The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obt…

Asymptotical stability of almost periodic solution for an impulsive multispecies competition-predation system with time delays on time scales

In this paper, we consider the almost periodic dynamics of an impulsive multispecies Lotka-Volterra competition system with time delays on time scales. By establishing some comparison theorems of dynamic equations with impulses and delays on time scales, a permanence result for the model is obtained. Furthermore, based on the permanence result, by studying the Lyapunov stability theory of impulsive dynamic equations on time scales, we establish a criterion for the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system. Finally, we give an examp…

A linearized stability theorem for nonlinear delay fractional differential equations

In this paper, we prove a theorem of linearized asymptotic stability for fractional differential equations with a time delay. More precisely, using the method of linearization of a nonlinear equation along an orbit (Lyapunov's first method), we show that an equilibrium of a nonlinear Caputo fractional differential equation with a time delay is asymptotically stable if its linearization at the equilibrium is asymptotically stable. Our approach based on a technique which converts the linear part of the equation into a diagonal one. Then using properties of generalized Mittag-Leffler functions, t…

Differentiability of Palmer's linearization Theorem and converse result for density functions

We study differentiability properties in a particular case of the Palmer's linearization Theorem, which states the existence of an homeomorphism $H$ between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that $H$ is a $C^{2}$ preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a nonlinear system in the nonautonomous case.

Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays

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…

带传输与反应延迟的渐近共识：综述

Asymptotic consensus with transmission and reaction delay: an overview

本文综述了带传输延迟与反应延迟的Hegselmann-Krause型模型的渐近共识结果，指出传输延迟无需小性假设即可达成共识，而反应延迟需足够小，并给出四种情形下的共识定理及相应证明方法。

Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction

We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this equation possesses pushed traveling fronts. First we prove that the pushed wavefronts are nonlinearly stable with asymptotic phase. Moreover, combinations of these waves attract, uniformly on $R$, every solution of equation $(*)$ with the initial datum sufficiently rapidly decaying at one (or at the both) infinities of the real line. These results provide a sh…

Asymptotic and exponential decay in mean square for delay geometric Brownian motion

We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of asymptotic decay. For exponential decay, they are easily resolvable numerically. The analytical method is based on construction of a Lyapunov functional (asymptotic decay) and forward-backward estimate for the square mean (exponential decay).