Search

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]

This paper aims to construct an effective numerical scheme to solve convection-reaction-diffusion problems consisting of time-fractional derivative and delay in time. First, the semi-discretization process is given for the fractional derivative using a finite-difference scheme with second-order accuracy. Then the cubic B-spline collocation method is employed to get the full discretization. We prove that the suggested scheme is conditionally stable and convergent. Two numerical examples are incorporated to verify the effectiveness of the algorithm. Numerical investigations support the proposed method's accuracy and show that the method solves the problem efficiently. [ABSTRACT FROM AUTHOR]

This article compared two high-order numerical schemes for convection-diffusion delay differential equation via two fractional operators with singular kernels. The objective is to present two effective schemes that give (3-α)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(3-\alpha )$$\end{document} and second order of accuracy in the time direction when α∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1)$$\end{document} using Caputo and Modified Atangana-Baleanu Caputo derivatives, respectively. We also implemented a trigonometric spline technique in the space direction, giving second order of accuracy. Moreover, meticulous analysis shows these numerical schemes to be unconditionally stable and convergent. The efficiency and reliability of these schemes are illustrated by numerical experiments. The tabulated results obtained from test examples have also shown the comparison of these operators. [ABSTRACT FROM AUTHOR]

We study a free boundary problem modeling multilayer tumor growth with a small time delay τ, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper, we establish the well‐posedness of the problem; namely, first, we prove that there exists a unique flat stationary solution (σ∗,p∗,ρ∗,ξ∗) for all μ>0. The stability of this stationary solution should depend on the tumor aggressiveness constant μ. It is also unrealistic to expect the perturbation to be flat. We show that, under non‐flat perturbations, there exists a threshold μ∗>0 such that (σ∗,p∗,ρ∗,ξ∗) is linearly stable if μ<μ∗ and linearly unstable if μ>μ∗. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications. [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]

The delayed logistic equation (also known as Hutchinson's equation or Wright's equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here, we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible non-trivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an L 2 -perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic-type models of growth with delays. [ABSTRACT FROM AUTHOR]

Real-time hybrid simulation evaluates the response of a structure in real time. In this study, a building with multi-story structure is divided into numerical and experimental substructures, and the vibration behavior of the experimental story is studied among the real-time simulation of the other stories. For applying the effect of static and inertial forces produced by the other stories to the experimental story, an electrohydraulic actuator and a shake table are used, respectively. The dynamics of the electrohydraulic actuator and the shake table can be estimated entirely by time delays, and these delays in the loop of simulation can reduce accuracy and cause system instability. Therefore, a delayed differential equation is used to determine the critical time delays depending on the system parameters. Results of simulation show the effect of non-dimensional parameters and time delays on the stability margin of hybrid simulation. [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]

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic equation. For the Hopf bifurcation analysis, the delay parameter is chosen as a bifurcation parameter. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. (1981) [7]. Furthermore, the direction of the bifurcation, the stability and the period of periodic solutions are given. Finally, the theoretical results are supported by some numerical simulations. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate stability conditions of equilibrium points of a general delay difference population model with and without Allee effects which occur at low population density. The analysis demonstrates that Allee effects have both stabilizing and destabilizing effects on population dynamics including time delay. [Copyright &y& Elsevier]

Abstract: This paper studies the exponential stability problem for a class of networked control systems (NCSs) with time delays and packet dropouts. The state-feedback closed-loop NCS is modeled as a new discrete-time switched system. Through using the parameter-dependent Lyapunov function, a sufficient condition is obtained for the exponential stability of the NCS under a packet-dropout rate. A linear matrix inequality approach is employed to design the state-feedback controller. An illustrative example is presented to demonstrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

In this paper, a model of predator-prey with disease in food chain is investigated - where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a differential–algebraic biological economic system with time delay where the model with Holling type II functional response incorporates a constant prey refuge and prey harvesting. By considering time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the differential–algebraic biological economic system based on the new normal form approach of the differential–algebraic system and the normal form approach and the center manifold theory. Finally, numerical simulations illustrate the effectiveness of our results. [Copyright &y& Elsevier]

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous. [ABSTRACT FROM AUTHOR]

Abstract: By introducing discrete time delay into the model for producing 1,3-propanediol by microbial continuous fermentation, we consider the stability and Hopf bifurcation of the delay differential system. Through numerical simulations, we get the rule of branch value changing with parameter and draw the pictures of periodic solutions and phase diagrams with specified parameters. The effect of time delay suggests that the system can qualitatively describe oscillatory phenomena occurring in the experiment. [Copyright &y& Elsevier]

In this survey, we briefly review some of our recent studies on predator-prey models with discrete delay. We first study the distribution of zeros of a second degree transcendental polynomial. Then we apply the general results on the distribution of zeros of the second degree transcendental polynomial to various predator-prey models with discrete delay, including Kolmogorov-type predator-prey models, generalized Gause-type predator-prey models with harvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey models with nonmonotone functional response and in delayed predator-prey model with predator harvesting are also introduced. [ABSTRACT FROM AUTHOR]

Energy efficient contact less suspension of high-speed rotor shafts in application areas such as power-plants and Flywheel Energy Storage Systems (FESS), can be achieved through Active Magnetic Bearings (AMBs). Inevitably, there exists time delay in sensing the rotor displacement and actual application of control force by electromagnets, thus leading to system instability. This paper therefore analyses stability of AMBs, with constant time delay. FESS is modelled as a rigid rotor-AMB system. Performance of two different control laws, (i) PID and (ii) a recently introduced Four-Element (FE) law, are compared. The governing equations are transformed to frequency domain, to exploit Pade's approximation in modeling time delayed value of feedback signal and stability analysis is carried out by analyzing the system poles. This methodology has been verified using experimental results from the existing literature. Numerical simulation of delay differential equations is also used to validate analytical results. The FE control law shows promise in tolerating delay, as compared to conventional PID controller thus leading to AMB system with higher possible spin speed and relaxed sampling constraints on AMB system instrumentation. One of the prospective applications of this work is in design of flywheel energy storage systems for an electric vehicle. • Time delay in Active Magnetic Bearing (AMB) system is inevitable. • Delayed control force from AMB on rotor may cause instability. • Stability analysis of 4-DOF rotor AMB system with constant time delay is reported. • For the first time, such analysis is done with AMB using integral control action. • Verification with experimental results and numerical simulation is carried out. [ABSTRACT FROM AUTHOR]

We consider the spatio-temporal patterns of disease spread involving structured populations. We start with a general model framework in population biology and spatial ecology where the individual's spatial movement behaviors depend on its maturation status, and we show how delayed reaction diffusion equations with nonlocal interactions arise naturally. We then consider the impact of this delayed nonlocal interaction on the disease spread by revisiting the spatial spread of rabies in continental Europe during the period between 1945 and 1985. We show how the distinction of territorial patterns between juvenile and adult foxes, the main carriers of the rabies under consideration, yields a class of partial differential equations involving delayed and nonlocal terms that are implicitly defined by a hyperbolic-parabolic equation, and we show how incorporating this distinction into the model leads to a formula describing the relation of the minimal wave speed and the maturation time of foxes. We show how the homotopy argument developed by Chow, Lin, and Mallet-Paret can be applied to obtain the existence of a heteroclinic orbit between a disease-free equilibrium and an endemic state for the spatially averaged system of delay differential equations, and we illustrate how the technique developed by Faria, Huang, and Wu can be used to establish the existence of a family of traveling wavefronts in the neighborhood of the heteroclinic orbit for the corresponding spatial model. [ABSTRACT FROM AUTHOR]

The initial value problem for the third order delay differential equation in a Hilbert space with an unbounded operator is investigated. The absolute stable three-step difference scheme of a first order of accuracy is constructed and analyzed. This difference scheme is built on the Taylor's decomposition method on three and two points. The theorem on the stability of the presented difference scheme is proven. In practice, stability estimates for the solutions of three-step difference schemes for different types of delay partial differential equations are obtained. Finally, in order to ensure the coincidence between experimental and theoretical results and to clarify how efficient the proposed scheme is, some numerical experiments are tested. [ABSTRACT FROM AUTHOR]