In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.
We present a condition for delay-independent stability of a class of nonlinear positive systems. This result applies to systems that are not necessarily monotone and extends recent work on cooperative nonlinear systems.
This document states the normal vector system for modified Hopf boundaries of delay differential systems with state and parameter dependent delays. Specifically, it states the proof for Proposition 1 in the paper entitled "Robust optimization of delay differential equations with state and parameter dependent delays" by the same authors [1].
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 …
In this paper, we propose a delayed perturbation of Mittag-Leffler type matrix function, which is an extension of the classical Mittag-Leffler type matrix function and delayed Mittag-Leffler type matrix function. With the help of the delayed perturbation of Mittag-Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations.
Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form 1/n, this study finds the asymptotic shape and size of the stability region. For example, a delay ration of 1/3 asymptotically produces a stability region 44.3% larger than any nearby delay ratios, showing extreme sensitivity in the delays. The study provides a systematic and geometric approach to finding the eigenvalues on the boundary of stability for this delay diffe…
We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.
The paper deals with the global asymptotic stability of general nonlinear time-delay systems with delay-dependent impulses through the Lyapunov-Krasovskii method. We derive a unified stability criterion which can be applied to a variety of impulsive systems. The cases when each of the continuous dynamics and the impulsive component is either stabilizing or destabilizing are investigated. Both theoretically and numerically, we demonstrate that the obtained result is more general than those existing in the literature.
We propose a delayed Mittag-Leffler type matrix function with logarithm, which is an extension of the classical Mittag-Leffler type matrix function with logarithm and delayed Mittag-Leffler type matrix function. With the help of the delayed Mittag-Leffler type matrix function with logarithm, we give an explicit form a of solutions to nonhomogeneous Hadamard type fractional time-delay linear differential equations. Moreover, we study existence uniqueness and stability in Ulam-Hyers sense of the Hadamard type fractional time-delay nonlinear equations.
We prove that characteristic equations of certain types of delay differential systems, under some mild conditions on their coefficients, can possess infinitely many complex roots.
In this paper, an original result in terms of a sufficient condition to test identifiability of nonlinear delayed-differential models with constant delays and multi-inputs is given. The identifiability is studied for the linearized system and a criterion for linear systems with constant delays is provided, from which the identifiability of the original nonlinear system can be proved. This result is obtained by combining a classical identifiability result for nonlinear ordinary differential systems due to M.S. Grewal and K. Glover (1976) with the identifiability of linear delayed-differential m…
Delay mass-action systems provide a model of chemical kinetics when past states influence the current dynamics. In this work, we provide a graph-theoretic condition for delay stability, i.e., linear stability independent of both rate constants and delay parameters. In particular, the result applies when the system has no delay, implying asymptotic stability for the ODE system. The graph-theoretic condition is about cycles in the directed species-reaction graph of the network, which encodes how different species in the system interact.
In this paper, the stability analysis of quaternion-valued neural networks (QVNNs) with both leakage delay and additive time-varying delays is proposed. By employing the Lyapunov-Krasovskii functional method and fully considering the relationship between time-varying delays and upper bounds of delays, some sufficient criteria are derived based on reciprocally convex method and several inequality techniques. The stability criteria are established in two forms: quaternion-valued linear matrix inequalities (QVLMIs) and complex-valued linear matrix inequalities (CVLMIs), in which CVLMIs can be dir…
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.
This paper is concerned with the traveling wave solutions and asymptotic spreading of delayed lattice differential equations without quasimonotonicity. The spreading speed is obtained by constructing auxiliary equations and using the theory of lattice differential equations without time delay. The minimal wave speed of invasion traveling wave solutions is established by presenting the existence and nonexistence of traveling wave solutions.
The paper discusses linear fractional representations of parameter-dependent nonlinear systems with dynamics defined by real rational nonlinearities and a finite set of point delays. The global asymptotic stability is investigated via linear matrix inequalities.
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…
We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a nonnegative coefficient. Some examples on nonstandard time scales are also given to show applicability and sharpness of the new results.
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…
In this paper a stability analysis for a Cournot duopoly model with tax evasion and time-delay in a continuous-time framework is presented. The mathematical model under consideration follows a gradient dynamics approach, is nonlinear and four-dimensional with state variables given by the production and declared revenue of each competitor. We prove that both the marginal cost rate and time delay play roles as bifurcation parameters. More precisely, if the marginal cost rate lies in certain closed interval then the equilibrium point is delay-independent stable, otherwise it is delay-dependent st…
Proportional delay is a particular case of time dependent delay. In this article, we consider differential equations involving multiple delays. The series solution of this equation leads to a class of special functions. This class of special functions is independent from all the existing special functions obtained as a solution of differential equations. We analyze the basic properties of this class and discuss various identities and relations.
Some errors contained in the author's previous article "An example of Bautin-type bifurcation in a delay differential equation", JMAA, 329(2007), 777-789, are listed and corrected. Original abstract: In a previous paper we gave sufficient conditions for a system of delay differential equations to present Bautin-type bifurcation. In the present work we present an example of delay equation that satisfies these conditions.
The main purpose of this paper is to provide a summary of the fundamental methods for analyzing delay differential equations arising in biology and medicine. These methods are employed to illustrate the effects of time delay on the behavior of solutions, which include destabilization of steady states, periodic and oscillatory solutions, bifurcations, and stability switches. The biological interpretations of delay effects are briefly discussed.
In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay.
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.
Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.
We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions by transforming the system of state-dependent delay equations into an abstract ordinary differential equation in a subspace of the sequence space $l^{\infty}(\mathbb{R}^{N+1})$ and prove the existence of complex extension of the bounded solutions. An example is given to illustrate the general results.
We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$ \dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $$ where $$ |a(t)|<1,~ b(t)\geq 0, ~h(t)\leq t, ~g(t)\leq t, $$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.
We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the $S^1$-equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations.
This paper analyzes the eigenvalue distribution of neutral differential systems and the corresponding difference systems, and establishes the relationship between the eigenvalue distribution and delay-independent stability of neutral differential systems. By using the ``Complete Discrimination System for Polynomials", easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems are established. The algebraic criteria generalize and unify the relevant results in the literature. Moreover, the maximal delay bound guaranteei…
We introduce a delay nonlinear differential equation model which describes how fish are harvested. In our previous studies we investigated the persistence of that equation and existence of a periodic solution for this equation. Here we study the stability (local and global) of the periodic solutions of that equation.
The $μ$-neutral linear fractional multi-delayed differential nonhomogeneous system with noncommutative coefficient matrices is introduced. The novel $μ$-neutral multi-delayed perturbation of Mittag-Leffler type matrix function is proposed. Based on this, an explicit solution to the system is investigated step by step. The existence uniqueness of solutions to $μ$-neutral nonlinear fractional multi-delayed differential system is obtained with regard to the supremum norm. The notion of stability analysis in the sense of solutions to the described system is discussed on the grounds of the fixed po…
The problem considered in this paper consists of a cascade of reactions with discrete as well as distributed delays, which arose in the context of Hes1 gene expression. For the abstract general model sufficient conditions for global stability are presented. Then the abstract result is applied to the Hes1 model.
We study the two state model which describes the balance equation for carbon dioxide and oxygen. These are nonlinear parameter dependent and because of the transport delay in the respiratory control system, they are modeled with delay differential equation. So, the dynamics of a two state one delay model are investigated. By choosing the delay as a parameter, the stability and Hopf bifurcation conditions are obtained. We notice that as the delay passes through its critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. The stable region of the system with dela…
A proper discretization of the logistic differential equation, which is preserving these two distinct equilibrium solutions and their unstability and stability, suggest that we need to examine the time delay of the logistic map. According to Murray, the effect of delay in models is "usually" to increase the potential for instability. However the word "usually" is really ambiguous. In this paper, we mathematically formulate and prove the two conjectures about stability and instability.
For systems of delay differential equations the Hopf bifurcation was investigated by several authors. The problem we consider here is that of the possibility of emergence of a codimension two bifurcation, namely the Bautin bifurcation, for some of such systems.
We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay system $D_{0,t}^{μ,ν}z\left( t\right) +Az\left( t\right) +Ωz\left( t-h\right) =f\left( t\right) $ of order $1<μ<2$ and type $0\leqν\leq1,$ with nonpermutable matrices $A$ and $Ω$. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear tim…
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 $x'(t)=\int_0^tk(t,s)h(x(s))ds$
This note is concerned with stability analysis of integral delay systems with multiple delays. To study this problem, the well-known Jensen inequality is generalized to the case of multiple terms by introducing an individual slack weighting matrix for each term, which can be optimized to reduce the conservatism. With the help of the multiple Jensen inequalities and by developing a novel linearizing technique, two novel Lyapunov functional based approaches are established to obtain sufficient stability conditions expressed by linear matrix inequalities (LMIs). It is shown that these new conditi…
Stability analysis of Richardson models with delay for confrontation between two countries
Departamento de Ciências Matemáticas and CEMS.UL, Faculdade de Ciências, Universidade de Lisboa（里斯本大学理学院数学科学系和CEMS.UL）；S.P.Timoshenko Institute of Mechanics of National Academy of Sciences of Ukraine（乌克兰国家科学院S.P.蒂莫申科力学研究所）
In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the C-norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone.
We employ the Craig-Wayne-Bourgain method to construct quasi-periodic solutions for delayed perturbation differential equations. Our results not only implement the existing literatures on constructing quasi-periodic solutions for DDE by the KAM method and space splitting technique, but also provide an example of application of multi-scale analysis method to non-selfadjoint problem.
We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small.
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.
This paper studies the link between the number of critical eigenvalues and the number of delays in certain classes of delay-differential equations. There are two main results. The first states that for k purely imaginary numbers which are linearly independent over the rationals, there exists a scalar delay-differential equation depending on k fixed delays whose spectrum contains those k purely imaginary numbers. The second result is a generalization of the first result for delay-differential equations which admit a characteristic equation consisting of a product of s factors of scalar type. In…
An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions …
Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.
We study the stability of general $n$-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential asymptotic stability of the system are given. In the second case, the stability depends on the size of some bounded diagonal delays and coefficients, although terms with unbounded delay may co-e…
Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.
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.
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.
