Search

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature. [ABSTRACT FROM AUTHOR]

We present a new method for obtaining stability conditions for certain classes of delay differential equations. The method is based on the transition from an individual equation to a family of equations, and next the selection of a representative of this family, the test equation, asymptotic properties of which determine those of all equations in the family. This approach allows us to obtain the conditions that are the criteria for the stability of all equations of a given family. These conditions are formulated in terms of the parameters of the class of equations being studied, and are effectively verifiable. The main difference of the proposed method from the known general methods (using Lyapunov–Krasovsky functionals, Razumikhin functions, and Azbelev W-substitution) is the emphasis on the exactness of the result; the difference from the known exact methods is a significant expansion of the range of applicability. The method provides an algorithm for checking stability conditions, which is carried out in a finite number of operations and allows the use of numerical methods. [ABSTRACT FROM AUTHOR]

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields. [ABSTRACT FROM AUTHOR]

Forest pests and diseases have been seriously threatening ecological security. Effective prevention and control of such threats can extend the growth cycle of forest trees and increase the amount of forest carbon sink, which makes a contribution to achieving China's goal of "emission peak and carbon neutrality". In this paper, based on the insect-vector populations (this refers to Monochamus alternatus, which is the main vector in Asia) in pine wilt disease, we establish a two-dimensional delay differential equation model to investigate disease control and the impact of time delay on the effectiveness of it. Then, we analyze the existence and stability of the equilibrium of the system and the existence of Hopf bifurcation, derive the normal form of Hopf bifurcation by using a multiple time scales method, and conduct numerical simulations with realistic parameters to verify the correctness of the theoretical analysis. Eventually, according to theoretical analysis and numerical simulations, some specific suggestions are put forward for prevention and control of pine wilt disease. [ABSTRACT FROM AUTHOR]

We discuss an analytical method for qualitative investigations of linear fractional delay differential equations. This method originates from the Lambert function technique that is traditionally used in stability analysis of ordinary delay differential equations. Contrary to the existing results based on such a technique, we show that the method can result into fully explicit stability criteria for a linear fractional delay differential equation, supported by a precise description of its asymptotics. As a by-product of our investigations, we also state alternate proofs of some classical assertions that are given in a more lucid form compared to the existing proofs. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems. [ABSTRACT FROM AUTHOR]

• Considering the randomness of the equivalent conicity of the wheelset system. • The analysis of wheelset system under non-smooth condition is more appropriate. • Considering the time delay displacement feedback control in the primary suspension. • Stochastic D(P)-bifurcation occur in the probabilistic sense in the wheelset system. • The hunting stability of the wheelset system during operation were analyzed. Considering not only the stochastic track irregularity and the possible effect of stochastic parameter excitation, but also the time delay of spring, that is, its response is in place, but the force generated is not in place, a stochastic delay wheelset system is established. The infinite dimensional system is reduced to the finite dimensional stochastic differential equation by using the center manifold, and further reduced to a one-dimensional diffusion process by using the stochastic averaging method. The stability of the wheelset system is obtained by analyzing the singular boundary theory and calculating the maximum Lyapunov exponent. The conditions and types of stochastic bifurcation in wheelset system are obtained by combining probability density function. The numerical simulation verifies the correctness of the theoretical analysis and shows that the time delay affects the critical hunting instability speed of wheelset. The stochastic term affects the lateral displacement of the stochastic delay wheelset system. [ABSTRACT FROM AUTHOR]

This article examines the stability problem of systems with interval time-varying delays. In the derivation of Lyapunov–Krasovskii functional (LKF), non-convex higher-degree polynomials may arise with respect to interval time-varying delays, making it difficult to determine the negative definiteness of LKF's derivative. This study was conducted to obtain stability conditions that can be described as linear matrix inequalities (LMIs). By considering the idea of matrix transition and introducing the delay-dependent augmented vector, a novel higher-degree polynomial inequality is proposed under the condition that the lower bound of the polynomial function variable is non-zero, which encompasses the existing lemmas as its special cases. Then, benefiting from this inequality, a stability criterion is derived in terms of LMIs. Finally, several typical examples are presented to verify the availability and strength of the stability condition. [ABSTRACT FROM AUTHOR]

We consider an initial problem (IP) for a linear neutral system with distributed delays and derivatives in Caputo's sense of incommensurate order, with different kinds of initial functions. In the case when the initial functions are with bounded variation, it is proven that this IP has a unique solution. The Krasnoselskii's fixed point theorem, a very appropriate tool, is used to prove the existence of solutions in the case of the neutral systems. As a corollary of this result, we obtain the existence and uniqueness of a fundamental matrix for the homogeneous system. In the general case, without additional assumptions of boundedness type, it is established that the existence and uniqueness of a fundamental matrix lead existence and uniqueness of a resolvent kernel and vice versa. Furthermore, an explicit formula describing the relationship between the fundamental matrix and the resolvent kernel is proven in the general case too. On the base of the existence and uniqueness of a resolvent kernel, necessary and sufficient conditions for the stability of the zero solution of the homogeneous system are established. Finally, it is considered a well-known economics model to describe the dynamics of the wealth of nations and comment on the possibilities of application of the obtained results for the considered systems, which include as partial case the considered model. [ABSTRACT FROM AUTHOR]

A class of nonlinear simple form differential delay equations with a τ-periodic coefficient and a constant delay τ > 0 is considered. It is shown that for an arbitrary value of the period T > 4τ -- d0, for some d0 > 0, there is an equation in the class such that it possesses an asymptotically stable T-period solution. The periodic solutions are constructed explicitly for the piecewise constant nonlinearities and the periodic coefficients involved, by reduction of the problem to one-dimensional maps. The periodic solutions and their stability properties are shown to persist when the nonlinearities are "smoothed" at the discontinuity points. [ABSTRACT FROM AUTHOR]

Many ecological models have received much attention in the past few years. In particular, predator–prey interactions have been examined from many angles to capture and explain various environmental phenomena meaningfully. Although the consumption of prey directly by the predator is a well-known ecological phenomenon, theoretical biologists suggest that the impact of anti-predator behavior due to the fear of predators (felt by prey) can be even more crucial in shaping prey demography. In this article, we develop a predator–prey model that considers the effects of fear on prey reproduction and on environmental carrying capacity of prey species. We also include two delays: prey species birth delay influenced by fear of the predator and predator gestation delay. The global stability of each equilibrium point and its basic dynamical features have been investigated. Furthermore, the "paradox of enrichment" is shown to exist in our system. By analysing our system of nonlinear delay differential equations, we gain some insights into how fear and delays affect on population dynamics. To demonstrate our findings, we also perform some numerical computations and simulations. Finally, to evaluate the influence of a fluctuating environment, we compare our proposed system to a stochastic model with Gaussian white noise terms. [ABSTRACT FROM AUTHOR]

The goal of this study is to derive a class of random impulsive fractional stochastic differential equations with finite delay that are of Caputo-type. Through certain constraints, the existence of the mild solution of the aforementioned system are acquired by Kransnoselskii's fixed point theorem. Furthermore, through Ito isometry and Gronwall's inequality, the Hyers-Ulam stability of the reckoned system is evaluated using Lipschitz condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating one type of nonlinear two-term fractional order delayed differential equations involving Caputo fractional derivatives. The Leray–Schauder alternative fixed-point theorem and Banach contraction principle are applied to analyze the existence and uniqueness of solutions to the problem with infinite delay. Additionally, the Hyers–Ulam stability of fractional differential equations is considered for the delay conditions. [ABSTRACT FROM AUTHOR]

• A general non-local delay model was developed by age-infected law. • Global stability and uniformly persistence are studied. • Mathematical result support that viruses therapy can decrease the tumor load. • Bayesian information criterion was adopted to select a better model when fitting the experimental data. The oncolytic virus is regarded as a novel, powerful, and biologically safe method of cancer treatment. A general delay differential system was driven by the age-dependent model better to understand the interaction between tumor cells and viruses. General continuous functions F (x , y) and G (x) depict the tumor proliferation rate and virus infection rate. The critical threshold value R 0 was calculated that plays a determinant role in whether virus therapy occurs. The non-local delay term ∫ t − τ t β G (x (θ)) v (θ) e − α (t − θ) d θ makes our model hard to analyze when using the traditional eigenvalue method. The method combining implicit function theorem and comparison theorem is used to overcome this problem. Furthermore, we support the fact that virotherapy can lead to tumor remission by using the fluctuation method. Lastly, Bayesian information criterion was adopted to select a better model when fitting the experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we propose numerical methods for the Riesz space distributed-order advection–diffusion equation with delay in 2D. We utilize the fractional backward differential formula method of second order (FBDF2), and weighted and shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and develop the finite difference method for the RFADED. It has been shown that the obtained schemes are unconditionally stable and convergent with the accuracy of O(h2+k2+κ2+σ2+ρ2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{O}({h^2} + {k^2} +{\kappa ^2} + {\sigma ^2} + {\rho ^2})$$\end{document}, where h, k and κ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\kappa$$\end{document} are space step for x, y and time step, respectively. Also, numerical examples are constructed to demonstrate the effectiveness of the numerical methods, and the results are found to be in excellent agreement with analytic exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability and oscillation of fractional differential equations cDαx(t) + ax(t) + ∫0¹ x(s + [t - 1])dR(s) = 0. We discretize the fractional differential equation by variation of constant formula and semigroup property of Mittag-Leffler function, and get the difference equation corresponding to the integer points. From the equivalence analogy of qualitative properties between the difference equations and the original fractional differential equations, the necessary and sufficient conditions of oscillation, stability and exponential stability of the equations are obtained. [ABSTRACT FROM AUTHOR]

• We proposed a mathematical pathogen-host model with a time delay for the dynamics of the banana black leaf streak disease. • Model accounted for the two reproduction means of the pathogen spores, seasonality and time delay to describe the incubation. • The basic reproduction number R0 does not depend on the time delay and is related to both sexual and asexual spore production. • The stability of the system was shown to not depend on the time delay, i.e. on the duration of the incubation period. • Results proved that the control of sexual spore production is not sufficient. We provide numerical simulations. Black Sigatoka Disease, also called Black Leaf Streak Disease (BLSD), is caused by the fungus Mycosphaerella fijiensis and is arguably one of the most important pathogens affecting the banana and plantain industries. Theoretical results on its dynamics are rare, even though theoretical descriptions of epidemics of plant diseases are valuable steps toward their efficient management. In this paper, we propose a mathematical model describing the dynamics of BLSD on banana or plantain leaves within a whole field of plants. The model consists of a system of periodic non-autonomous differential equations with a time delay that accounts for the time of incubation of M. fijiensis ' spores. We compute the basic reproduction number of the disease and show that it does not depend on the time delay, meaning that the persistence of BLSD would not qualitatively change even if the incubation period of the pathogen is perturbed. We derive local and global long-term dynamics of the disease and provide numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness and existence of the solution of a non-autonomous and nonsingular delay difference equation using the well-known principle of contraction from fixed point theory. Furthermore, we study the Hyers–Ulam stability of the given system on a bounded discrete interval and then on an unbounded interval. An example is also given at the end to illustrate the theoretical work. [ABSTRACT FROM AUTHOR]

Chatter is not conducive to machining efficiency and surface quality. One of the essential types of chatter in the machining process is regenerative chatter. This study presents the numerical integration scheme–based semi-discretization methods (NISDMs) for milling stability prediction. Firstly, the dynamic model of the milling process is represented by the delay differential equation (DDE). The forced vibration period is discretized into many small-time intervals. After integrating on the small-time interval, only the time-delay term–related part is approximated by different order numerical integration schemes. Both the free and forced vibration processes are considered in the derivation process. The state transition matrix is constructed by mapping the dynamic displacement between the current and previous time periods. The NISDMs are compared with the benchmark methods in terms of the rate of convergence and computational time. The comparison results show that the NISDMs converge faster than the benchmark methods. To improve the computational efficiency of the NISDMs, the precise integration method is used in the calculation process. The computational time consumed by the NISDMs is much less than that consumed by the benchmark methods. The NISDMs are proved to be more accurate and efficient methods for stability prediction in milling than the other considered methods. [ABSTRACT FROM AUTHOR]

A swarming model is a model that describes the behavior of the social aggregation of a large group of animals or the community of humans. In this work, the swarming model that includes the short-range repulsion and long-range attraction with the presence of time delay is investigated. Moreover, the convergence to a consensus representing dispersion and cohesion properties is proved by using the Lyapunov functional approach. Finally, numerical results are provided to demonstrate the effect of time delay on the motion of the group of agents. [ABSTRACT FROM AUTHOR]

This paper proposes a coupled-circuit system composed of two Chua's circuits with lossless transmission lines. By applying the Kirchhoff's voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point. [ABSTRACT FROM AUTHOR]

To reduce the global burden of mosquito-borne diseases, e.g. dengue, malaria, the need to develop new control methods is to be highlighted. The sterile insect technique (SIT) and various genetic modification strategies, have a potential to contribute to a reversal of the current alarming disease trends. In our previous work, the ordinary differential equation (ODE) models with different releasing sterile mosquito strategies are investigated. However, in reality, implementing SIT and the releasing processes of sterile mosquitos are very complex. In particular, the delay phenomena always occur. To achieve suppression of wild mosquito populations, in this paper, we reassess the effect of the delayed releasing of sterile mosquitos on the suppression of interactive mosquito populations. We extend the previous ODE models to the delayed releasing models in two different ways of releasing sterile mosquitos, where both constant and exponentially distributed delays are considered, respectively. By applying the theory and methods of delay differential equations, the effect of time delays on the stability of equilibria in the system is rigorously analyzed. Some sustained oscillation phenomena via Hopf bifurcations in the system are observed. Numerical examples demonstrate rich dynamical features of the proposed models. Based on the obtained results, we also suggest some new releasing strategies for sterile mosquito populations. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of traveling waves of infinite-dimensional lattice differential equations with time delay, where the equation may be not quasi-monotone. Firstly, by applying Schauder's fixed point theorem, we get the existence of traveling waves with the speed c > c∗ (here c∗ is the minimal wave speed). Using a limiting argument, the existence of traveling waves with wave speed c = c∗ is also established. Secondly, for sufficiently small initial perturbations, the asymptotic stability of the traveling waves Φ : = { Φ (n + ct) } n ∈ ℤ with the wave speed c > c∗ is proved. Here we emphasize that the traveling waves Φ : = { Φ (n + ct) } n ∈ ℤ may be non-monotone. [ABSTRACT FROM AUTHOR]

By developing new efficient techniques and using an appropriate fixed point theorem, we derive several new sufficient conditions for the pseudo almost periodic solutions with double measure for some system of differential equations with delays. As an application, we consider certain models for neural networks with delays. [ABSTRACT FROM AUTHOR]

Stability prediction with both high computational accuracy and speed is still a challenging issue and has been attracting significant attention from the academia and industry. This study presents a Legendre-Chebyshev-based stability analysis method (LCM) for milling operations. According to the cutting state, it divides the system period of milling model into the free and the forced vibration time periods. By introducing appropriate transformation, the latter time interval is further discretized nonuniformly into the Chebyshev-Gauss-Lobatto points, which has explicit expression. Then, the state term over the discrete time points is approximated with the Legendre expansion, and its corresponding derivative is acquired via a novel and efficient algorithm. Thereafter, Floquet matrix within the system period of milling model can be determined for predicting the system stability via the known Floquet theory. Finally, we validate the effectiveness of the LCM by employing the single and two degrees of freedom (DOF) milling operations and making detailed comparisons with the recent representative algorithms, which indicates that the presented Legendre-Chebyshev-based method has both high prediction accuracy and speed. [ABSTRACT FROM AUTHOR]

The intention of this paper is to present an updated model to predict stability limits of five-axis ball-end milling. Two-degree of freedom five-axis ball-end milling system in consideration of regenerative effect and helix angle is first concluded into a delay differential equation (DDE) with time-varying coefficients. As the time period being carved up evenly into a certain number of elements, the discrete map of system response is determined with the assistance of generalized precise integration method (GPIM). Then, in a single tooth passing period, the analytic cutter-workpiece engagement (CWE) is extracted by an intersection of spatial surface technique to ascertain the instantaneous cutting angles. Taking the advantage of these angles, the transition matrix denoting the given machining state is established to predict the process stability. The validity of the predictive model is verified in a five-axis CNC machine tool by close accordance with the experimental results. Lastly, a set of comparisons and discussions are developed to demonstrate the feature of this updated predictive model. [ABSTRACT FROM AUTHOR]

• A network epidemic model with nonlinear incidence rate and time delay is regarded. • Stability is proved in a strict mathematical way. • Four different control strategies are investigated and compared in this paper. • Numerical simulations provide a new insight into epidemic diffusion on complex networks. In this paper, we establish a susceptible-infected-susceptible (SIS) epidemic model with nonlinear incidence rate and time delay on complex networks. Firstly, according to the existence of a positive equilibrium point, we work out the threshold values R 0 and λ c of disease propagation. Secondly, we demonstrate the stabilities of the disease-free equilibrium point and the disease-spreading equilibrium point by constructing Lyapunov function and applying delay differential equations theorem. Thirdly, four different control strategies are investigated and compared, including uniform immunization control, acquaintance immunization control, active immunization control and optimal control. Finally, we perform representative numerical simulations to illustrate the theoretical results and further discover that the nonlinear incidence rate can more accurately reflect individual psychological activities when a certain disease outbreaks at a high level. [ABSTRACT FROM AUTHOR]

This paper considers the stabilization problem of two interconnected linear subsystems of differential equations with constant delay; one of the subsystems has an exponential factor in the right-hand side. Sufficient conditions for the stability of this system are established and then used for its stabilization. [ABSTRACT FROM AUTHOR]

Abstract There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we develop a homotopy continuation method to find the characteristic roots of delay differential equations with multiple delays. We introduce a homotopy parameter μ into the characteristic equation in such a way that for μ = 0 this equation contains only one exponential term (corresponding to the largest delay) and for μ = 1 the original characteristic equation is recovered. For μ = 0, all the characteristic roots can be expressed in terms of the Lambert W function. Pseudo-arclength continuation is then used to trace the roots as a function of μ. We demonstrate the method on several test cases. Cases where it may fail are also discussed. [ABSTRACT FROM AUTHOR]

We are concerned with 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 delay differential equations. Sufficient conditions for the exponential asymptotic stability of the linear system are established. By using this stability, the permanence of the perturbed nonlinear system is studied. Under more restrictive constraints on the coefficients, the system has a cooperative type behaviour, in which case explicit uniform lower and upper bounds for the solutions are obtained. As an illustration, the asymptotic behaviour of a non-autonomous Nicholson system with distributed delays is analysed. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature. [ABSTRACT FROM AUTHOR]

We consider the system of delay differential equations (DDE) representing the models containing two cells with time-delayed connections. We investigate global, local stability and the bifurcations of the trivial solution under some generic conditions on the Taylor coefficients of the DDE. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including pitchfork, transcritical and Hopf bifurcation) and Takens-Bogdanov bifurcation as a codimension two bifurcation. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. Finally, we show that the analytical results agree with numerical simulations. [ABSTRACT FROM AUTHOR]

We study mechanical vibrations in milling with non-uniform pitch and variable helix tools. The process is modeled by a periodic delay differential equation with distributed delay, which takes into account, for example, the nonlinear cutting force behavior, the effect of runout, and the exact delay distribution due to the unequally spaced flutes. We present a new method for the identification of the chatter stability lobes from the linearized system that is based on the multifrequency solution. We give detailed remarks on the truncation of the resulting infinite dimensional matrices and the efficient numerical implementation of the method. Cutting tests for steel milling with a customary end mill with non-uniform pitch and variable helix angle and a conventional end mill with uniform pitch and constant helix angle are performed. The numerical and experimental results coincide well. They reveal a significant increase of the limiting depth of cut for the variable helix tool compared to the conventional tool. Moreover, we show that in contrast to conventional tools, for non-uniform pitch and variable helix tools, an exact model with time-varying coefficients, nonlinear cutting force behavior, and runout is necessary for an accurate prediction of the stability lobes. [ABSTRACT FROM AUTHOR]

Based on the idea of collocation methods, a spline-based integration method for the stability prediction of systems governed by time-periodic delay differential equations (DDEs) is presented. A B-spline curve is adopted to approximate the solution of the DDEs using the direct integration technique. The stability of the dynamic system is then predicted using the Floquet theory based on the established state transition matrix. The proposed method can apply to time-periodic DDEs where the delay and the period are incommensurate, and the proposed method is extended to study the stability of milling processes as well. For stability analysis of variable pitch cutters, the proposed method use one or multiple B-spline curves to approximate the solution of the DDEs according to the relationship between the cutter angles and the radial immersion, which can describe exactly the free vibration of the cutter. Compared with the numerical integration method and the Chebyshev collocation method, the simulation results demonstrate that the proposed technique is more efficient and accurate for stability prediction at low radial immersion cases in milling processes with variable pitch cutters. [ABSTRACT FROM AUTHOR]

We consider a characteristic equation to analyze asymptotic stability of a scalar renewal equation, motivated by structured population dynamics models. The characteristic equation is given by 1 = ∨ 0∞ k(a)e-Μ ada, where k : R+ R can be decomposed into positive and negative parts. It is shown that if delayed negative feedback is characterized by a convex function, then all roots of the characteristic equation locate in the left half complex plane. [ABSTRACT FROM AUTHOR]

This paper deals with positivity and stability of linear implicit difference delay equations. Being different from the Lyapunov function approach commonly used in stability analysis, the method employed in this paper gives a way to solve the exponential stability of linear implicit difference equations with time-varying delay. By decomposition state-space and mathematical induction method, new necessary and sufficient conditions for positivity and stability of such systems are derived. Numerical examples are given to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true. [ABSTRACT FROM AUTHOR]

In this paper, a Leslie-type predator-prey model with ratio-dependent functional response and Allee effect on prey is considered. We first study the existence of the multiple positive equilibria and their stability. Then we investigate the effect of delay on the distribution of the roots of characteristic equation and obtain the conditions for the occurrence of simple-zero, double-zero and triple-zero singularities. The formulations for calculating the normal form of the triple-zero bifurcation of the delay differential equations are derived. We show that, under certain conditions on the parameters, the system exhibits homoclinic orbit, heteroclinic orbit and periodic orbit. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of linear impulsive delay differential equations with piecewise continuous argument. The conditions for stability of the exact solution are obtained. Under these conditions, it is proven that the θ -method preserves stability of the equations. Furthermore, stability of Runge–Kutta methods for this kind of equations is studied. Some numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

A substitution is introduced which exploits the parameters of the high-order delayed linear non-autonomous models and consequently allows the use of the M -matrix methods for the stability analysis. To demonstrate the efficacy of the proposed algorithm we obtain explicit and practical stability conditions for the second and third order non-autonomous delayed models. [ABSTRACT FROM AUTHOR]

The main aim of this paper is the investigation of the stability problem for ordinary delay differential equations. More precisely, we would like to study the following problem. Assume that for a continuous function a given delay differential equation is fulfilled only approximately. Is it true that in this case this function is close to an exact solution of this delay differential equation?. [ABSTRACT FROM AUTHOR]

The objective of this study was to get some sufficient conditions which guarantee the asymptotic stability and uniform boundedness of the null solution of some differential equations of third order with the variable delay. The most efficient tool for the study of the stability and boundedness of solutions of a given nonlinear differential equation is provided by Lyapunov theory. However the construction of such functions which are positive definite with corresponding negative definite derivatives is in general a difficult task, especially for higher-order differential equations with delay. Such functions and their time derivatives along the system under consideration must satisfy some fundamental inequalities. Here the Lyapunov second method or direct method is used as a basic tool. By defining an appropriate Lyapunov functional, we prove two new theorems on the asymptotic stability and uniform boundedness of the null solution of the considered equation. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of third order with a constant delay. We also give an example to illustrate the importance of the theoretical analysis in this work and to test the effectiveness of the method employed. [ABSTRACT FROM AUTHOR]

In this paper, we study both the oscillation and the stability of impulsive differential equations when not only the continuous argument but also the impulse condition involves delay. The results obtained in the present paper improve and generalize the main results of the key references in this subject. An illustrative example is also provided. [ABSTRACT FROM AUTHOR]