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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 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]

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 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]

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]

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]

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]

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]

This paper considers the problem of robust $H_{\infty}$ control for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. By using the linear matrix inequality (LMI) approach, a sufficient condition is presented for a prescribed uncertain singular system with time-delay to have generalized quadratic stability and $H_{\infty}$ performance. Furthermore, the design methods of state feedback controllers are considered such that the resulting closed-loop system has generalized quadratic stability with $H_{\infty}$ performance. By means of matrix inequalities, sufficient conditions are derived for the existence of memory-less and memorial static state feedback controllers. The controllers are obtained by the solutions of matrix inequalities. [ABSTRACT FROM AUTHOR]

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = a f\left(y(t-\tau)\right) - b y(t)$, where D is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay. [ABSTRACT FROM AUTHOR]

We propose a new method for modeling the well-known phenomenon of 'bursting behavior' in neuron systems by invoking delay equations. Namely, we consider a singularly perturbed nonlinear difference-differential equation with two delays describing the functioning of an isolated neuron. Under a suitable choice of parameters, we establish the existence of a stable periodic motion with any prescribed number of spikes on a closed time interval equal to the period length. [ABSTRACT FROM AUTHOR]

This paper investigates the stability problem of Markovian jump systems with multiple delay components and polytopic uncertainties. A new Lyapunov-Krasovskii functional is used for the stability analysis of Markovian jump systems with or without polytopic uncertainties. Two numerical examples are provided to demonstrate the applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

By using the variational Liapunov method, stability properties in terms of two measures for delay differential equations are discussed. In the case that the unperturbed systems are ordinary differential systems, employing auxiliary measure h*(t,x), criteria on nonuniform and uniform stability in terms of two measures for delay differential equations are established. [ABSTRACT FROM AUTHOR]

In this paper, by using Liapunov functional, some sufficient conditions are obtained for the stability of the equilibrium of a neural network model with delay of the type [ABSTRACT FROM AUTHOR]

Alcoholism is a social phenomenon that affects all social classes and is a chronic disorder that causes the person to drink uncontrollably, which can bring a series of social problems. With this motivation, a delayed drinking model including five subclasses is proposed in this paper. By employing the method of characteristic eigenvalue and taking the temporary immunity delay for alcoholics under treatment as a bifurcation parameter, a threshold value of the time delay for the local stability of drinking-present equilibrium and the existence of Hopf bifurcation are found. Then the length of delay has been estimated to preserve stability using the Nyquist criterion. Moreover, optimal strategies to lower down the number of drinkers are proposed. Numerical simulations are presented to examine the correctness of the obtained results and the effects of some parameters on dynamics of the drinking model. [ABSTRACT FROM AUTHOR]

Analysis of systems involving delay is a popular topic among the applied scientists. In the present work, we analyse the generalised equation D α x (t) = g x (t - τ 1) , x (t - τ 2) involving two delays, viz. τ 1 ≥ 0 and τ 2 ≥ 0 . We use stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations. [ABSTRACT FROM AUTHOR]

Taking the delay due to the latent period of computer viruses and the delay due to the period that the anti-virus software uses to clean the computer viruses as the bifurcation parameters, local Hopf bifurcation of an epidemic model over the Internet is studied. We discuss the existence of the Hopf bifurcation under four conditions: (1) τ1>0, τ2=0, (2) τ1=0, τ2>0, (3) τ1=τ2=τ>0, and (4) τ1>0, τ2∈(0,τ20). Properties of the Hopf bifurcation about condition (4) are investigated by means of the center manifold theorem and the normal form theory. Finally, some simulations are presented to support our obtained results. [ABSTRACT FROM AUTHOR]