Search

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 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]

In this paper, the stability of an equilibrium of a feedback nonlinear system with time delay and structural switching is studied. The critical case of a zero root of the characteristic equations of the linearised systems is treated by applying a Malkin-type theorem using a complete Lyapunov–Krasovskii functional. [ABSTRACT FROM AUTHOR]

In this paper the stability of the zero equilibrium of a system with time delay is studied. The critical case of a multiple zero root of the characteristic equation of the linearized system is treated by applying a Malkin type theorem and using a complete Lyapunov-Krasovskii functional. An application to a model for malaria under treatment considering the action of the immune system is presented. [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, we study a fractional-order model with time-delay to describe the dynamics of Ebola virus infection with cytotoxic T-lymphocyte (CTL) response in vivo. The time-delay is introduced in the CTL response term to represent time required to stimulate the immune system. Based on fractional Laplace transform, some conditions on stability and Hopf bifurcation are derived for the model. The analysis shows that the fractional-order with time-delay can effectively enrich the dynamics and strengthen the stability condition of fractional-order infection model. Finally, the derived theoretical results are justified by some numerical simulations. [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]

The purpose of this paper is devoted to studying the implicit–explicit (IMEX) one-leg methods for stiff delay differential equations (DDEs) which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff DDEs are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods. [ABSTRACT FROM PUBLISHER]

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]

Abstract: Recently, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. In this paper, we present a delay differential model to describe the interactions between the effector and tumour cells. The existence of the possible steady states and their local stability and change of stability via Hopf bifurcation are theoretically and numerically investigated. Parameter estimation problem for given real observations, using least squares approach, is studied. The global stability and sensitivity analysis are also numerically proved for the model. The stability and periodicity of the solutions may depend on the time-lag parameter. The model is qualitatively consistent with the experimental observations of immune-induced tumour dormancy. The model also predicts dormancy as a transient period of growth which necessarily results in either tumour elimination or tumour escape. [Copyright &y& Elsevier]

Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation. [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]

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]

Abstract: In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of . The function describing stability of HB is periodic with respect to . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data. [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: This paper is concerned with the stability of -dimensional stochastic differential delay systems with nonlinear impulsive effects. First, the equivalent relation between the solution of the -dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding -dimensional stochastic differential delay system without impulsive effects is established. Then, some stability criteria for the -dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained. Finally, the stability criteria are applied to uncertain impulsive stochastic neural networks with time-varying delay. The results show that, this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks. Some examples are also discussed to illustrate the effectiveness of our theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we construct a two-layer feedback neural network to theoretically investigate the influence of symmetry and time delays on patterned rhythmic behaviors. Firstly, linear stability of the model is investigated by analyzing the associated transcendental characteristic equation. Next, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delays of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Thirdly, based on the normal form approach and the center manifold theory, we derive the formula to determine the bifurcation direction and stability of Hopf bifurcating periodic solutions. Finally, some numerical examples and the corresponding numerical simulations are used to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates control for a class of nonlinear systems with time-varying delays and convex polytopic uncertainties. A new type of parameter-dependent Lyapunov–Krasovskii functional is introduced to derive delay-dependent sufficient conditions for the optimal control with exponential stability. All the conditions developed in this paper are formulated in terms of linear matrix inequalities. Finally, a numerical example shows the effectiveness of the proposed methodology. [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: The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours. [Copyright &y& Elsevier]

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]

Abstract: When modeling neural networks in a real world, not only diffusion effect and fuzziness cannot be avoided, but also self-inhibitions, interconnection weights, and inputs should vary as time varies. In this paper, we discuss the dynamical behaviors of delayed reaction–diffusion fuzzy cellular neural networks with varying periodic self-inhibitions, interconnection weights as well as inputs. By using Halanay’s delay differential inequality, -matrix theory and analytic methods, some new sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of the periodic solution, and the exponentially convergent rate index is also estimated. In particular, the traditional assumption on the differentiability of the time-varying delays is no longer needed. The methodology developed in this paper is shown to be simple and effective for the exponential periodicity and stability analysis of neural networks with time-varying delays. Two examples are given to show the usefulness of the obtained results that are less restrictive than recently known criteria. [Copyright &y& Elsevier]

Abstract: A sex-structured mathematical model for heterosexual transmission of HIV/AIDS with explicit incubation period is presented as a system of discrete delay differential equations. The epidemic threshold and equilibria for the model are determined and stabilities are examined. The disease-free equilibrium is shown to be locally and globally stable when the basic reproductive number is less than unity. We use the Lyapunov functional approach to show that the endemic equilibrium is locally asymptotically stable. Further comprehensive qualitative analysis of the model including persistence and permanence are investigated. [Copyright &y& Elsevier]

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]

Abstract: Stability of linear systems with uncertain bounded time-varying delays is studied under the assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the -norm of a certain integral operator. There exists a bound on this operator norm in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving a tight operator bound which is an increasing and continuous function of the delay derivative upper bound . For the new bound corresponds to the second case and improves the existing bound. As a result, for the first time, delay-derivative-dependent frequency domain and time domain stability criteria are derived for systems with the delay derivative greater than 1. [Copyright &y& Elsevier]

We consider networks of coupled scalar maps, with weighted connections which may include a time delay, and study the stability of equilibria with respect to the delays and connection structure. We prove that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. The stability region in the parameter plane shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, all bipartite graphs have an identical stability region, regardless of the delay or graph size, which is also the smallest stability region among those of all graphs. Furthermore, for certain parameter ranges, unstable (and possibly chaotic) maps can be stabilized via diffusive coupling with an odd time delay, provided that the network does not have a nontrivial and connected bipartite component. On the other hand, stabilization is not possible for even values of the delay or for bipartite networks. [ABSTRACT FROM AUTHOR]

Transition from excitability to asymptotic periodicity in an excitable system, modeled by the FitzHugh–Nagumo equations, with multiple time-delays, is analyzed. It is demonstrated that, for intermediate time-lags, the system has two coexisting attractors, a hyperbolic stable fixed point and a stable limit cycle. The fixed point is destabilized via subcritical Hopf bifurcation for much larger values of the time-lags. [ABSTRACT FROM AUTHOR]