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• 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]

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]

• 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 a social contagion model with group interactions on heterogeneous network, in which the group interactions are represented by incorporating higher-order terms. The dynamics of the proposed model are analyzed, revealing the effect of group interactions on the system dynamics. We derive a global asymptotically stability condition of the zero equilibrium point. If the parameters enhancement factor and transform probability meet the condition, the social contagion will eventually disappear. The bifurcation behavior arising from group interactions is investigated, when the enhancement factor exceeds a threshold, the system undergoes a backward bifurcation. This implies that R 0 < 1 does not guarantee the disappearance of social contagion, we also need to control the initial values of social contagion at a lower level. The optimal control strategies for the model are provided. Moreover, the numerical simulations validate the accuracy of the theoretical analysis. It is worth noting that the group interactions lead to the emergence of a bistable phenomenon, in which the social contagion will either fade away or persist eventually, depending on the initial values. • The simplicial SIS social contagion model with delay is proposed on heterogeneous networks. • The dynamics of the model are analyzed, revealing the effect of group interactions on the system dynamics. • An optimal control strategy is proposed for the model, incorporating group interactions. [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]

This paper considers the problems on the existence and uniqueness, the p th(p ≥ 1)-moment and the almost sure stability with a general decay for the global solution of stochastic delay differential equations with Markovian switching, when the drift term and the diffusion term satisfy the locally Lipschitz condition and the monotonicity condition. By using the Lyapunov function approach, the Barbalat Lemma and the nonnegative semi-martingale convergence theorem, some sufficient conditions are proposed to guarantee the existence and uniqueness as well as the stability with a general decay for the global solution of such equations. It is mentioned that, in this paper, the time-varying delay is a bounded measurable function. The derived stability results are more general, which not only include the exponential stability but also the polynomial stability as well as the logarithmic one. At last, two examples are given to show the effectiveness of the theoretical results obtained. [ABSTRACT FROM AUTHOR]

A stability theory of analytic and numerical solutions to linear impulsive delay differential equations(IDDEs) is established. The stability results in existing literature are extended to IDDEs with variable impulses. A convergent numerical process is proposed to calculate numerical solutions to IDDEs with variable impulses. Convergence and stability of the numerical solutions are studied in the paper. Numerical experiments are given in the end to confirm the conclusion. [ABSTRACT FROM AUTHOR]

• Stability properties of DEs with a general distributed delay are investigated. • Saddle-node and Hopf bifurcation curves are determined in a general framework. • Stability region of the equilibrium is fully characterized in the parameter plane. • Criticality of Hopf bifurcation is analyzed by the multiple times scales method. • Theoretical results are showcased in the framework of a simple neural model. For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural 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]

Abstract As a type of differential equation, uncertain delay differential equation is driven by Liu process. Stability in measure, stability in mean and stability in moment for uncertain delay differential equation have been proposed. This paper mainly gives a concept of stability in distribution, and proves a sufficient condition for uncertain delay differential equation being stable in distribution as a supplement. Moreover, this paper further discusses their relationships among stability in distribution, stability in measure, stability in mean and stability in moment. [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]

Sea lice infection is one of the major threats in the marine fishery, especially for farmed salmon. In this paper, we propose a mathematical model for the growth of sea lice with a three-stage structure: non-infectious immature, infectious immature and adults where the level of non-infectious immature development depends on the size of the adult population. We first describe the nonlinear dynamics by a system of partial differential equations, then, by mathematical techniques and an appropriate change of variables transform it into a system of delay differential equations with constant delay. We address the system threshold dynamics in the established model with respect to the adult reproduction number R s , including the global stability of the trivial steady state when R s < 1 , persistence and global attractivity of the unique positive steady state when R s > 1 . Numerical simulations are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p = 0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion. [ABSTRACT FROM AUTHOR]

In this paper, fractional backward differential formulas (FBDF) are presented for the numerical solution of fractional delay differential equations (FDDEs) of the form λ n 0 C D t α n y ( t ) + λ n − 1 0 C D t α n − 1 y ( t ) + ⋯ + λ 1 0 C D t α 1 y ( t ) + λ n + 1 y ( t − τ ) = f ( t ) , t ∈ [ 0 , T ] , where λ i ∈ R ( i = 1 , ⋯ , n + 1 ) , λ n + 1 ≠ 0 , 0 ⩽ α 1 < α 2 < ⋯ < α n < 1 , T > 0 , in Caputo sense. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Also we find the Green's functions for this equation corresponding to periodic/anti-periodic conditions in terms of the functions of Mittag Leffler type. Numerical tests are presented to confirm the strength of the approach under investigation. [ABSTRACT FROM AUTHOR]

The paper discusses stability and asymptotic properties of a fractional-order differential equation involving both delayed as well as non-delayed terms. As the main results, explicit necessary and sufficient conditions guaranteeing asymptotic stability of the zero solution are presented, including asymptotic formulae for all solutions. The studied equation represents a basic test equation for numerical analysis of delay differential equations of fractional type. Therefore, the knowledge of optimal stability conditions is crucial, among others, for numerical stability investigations of such equations. Theoretical conclusions are supported by comments and comparisons distinguishing behaviour of a fractional-order delay equation from its integer-order pattern. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of highly nonlinear hybrid stochastic differential delay equations (SDDEs). Different from the most existing papers, the time delay functions in the SDDEs are no longer required to be differentiable, not to mention their derivatives are less than 1. The generalized Hasminskii-type theorems are established for the existence and uniqueness of the global solutions. Comparing with the existing results, we show our new theorems are much more general and can be applied to a much wider class of highly nonlinear SDDEs. Further sufficient conditions are also obtained for the asymptotic boundedness and stability. [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 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]

In this paper, we give a detailed Hopf bifurcation analysis of a recurrent neural network system involving both discrete and distributed delays. Choosing the sum of the discrete delay terms as a bifurcation parameter the existence of Hopf bifurcation is demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are studied by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included. [ABSTRACT FROM AUTHOR]

In this paper, a structure-preserving model reduction approach for a class of delay differential equations is proposed. Benefits of this approach are, firstly, the fact that the delay nature of the system is preserved after reduction, secondly, that input–output stability properties are preserved and, thirdly, that a computable error bound reflecting the accuracy of the reduction is provided. These results are applicable to large-scale linear delay differential equations with constant delays, but also extensions to a class of nonlinear delay differential equations with time-varying delays are presented. The effectiveness of the results is evidenced by means of an illustrative example. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical solutions of nonlinear delay-differential–algebraic equations with proportional delay, which are transformed into nonlinear delay- differential–algebraic equation with constant delay through exponential transformation. Block boundary value methods are extended to solve this type of equation, and their unique solvability, convergence, and stability are rigorously proved, respectively. The computational effectiveness of the methods and correctness of the theoretical result are illustrated through numerical examples. [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: The aim of this work is to study the dynamical behavior of a virus dynamics model with general incidence rate and two delays. The first delay represents the time from the virus entry to the production of new viruses and the second delay corresponds to the time necessary for a newly produced virus to become infectious. Lyapunov functionals are constructed and LaSalle invariance principle for delay differential equations is used to establish the global asymptotic stability of the disease-free and the chronic infection equilibria. The results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number and under some assumptions on the general incidence function. Our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection. These results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models. [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: A Galerkin projection scheme to obtain low dimensional approximations of delay differential equations (DDEs) involving state-dependent delays is developed. The current scheme is an extension of a similar, recently proposed scheme for DDEs with constant delays in the publication by P. Wahi, A. Chatterjee 2005. The resulting ordinary differential equations (ODEs) from the Galerkin scheme are easier to integrate using commercial ODE solvers, and are amenable to stability and bifurcation analysis using standard techniques. First, the application of the formulation is demonstrated through a scalar delay differential equation, and the performance of the formulation is assessed. Next, the scheme is applied to a two degrees-of-freedom model describing the coupled axial and torsional vibrations of oil well drill-strings. In both cases, the Galerkin approximations show an excellent agreements with the direct numerical simulations of the original systems. [Copyright &y& Elsevier]

Abstract: In this paper, a class of neutral stochastic differential equation is studied. By using the fixed point method, some conditions are established for ensuring the equation is bounded, asymptotically stable and -stable in mean square. Then, with Borel–Cantelli lemma, several sufficient conditions for almost surely asymptotic stability and -stability are given. The conditions do not require the boundedness of delays, nor do they ask for the continuousness on the coefficient functions. The results of this paper are new and they extend and improve previously known results. [Copyright &y& Elsevier]

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 work, we present an implicit compact difference scheme for solving a class of neutral delay parabolic differential equations (NDPDEs). The unique solvability and unconditional stability of the scheme are proved. The temporal accuracy of the scheme is improved by using different Richardson extrapolation techniques for linear and nonlinear problems, and fourth-order accuracy in both temporal and spatial dimensions is obtained. Finally, numerical experiments are conducted to verify the accuracy and efficiency of the algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, the global stability problem for a general discrete Cohen–Grossberg neural network with finite and infinite delays is investigated. A simple criterion ensuring the global asymptotical stability is established, by applying the Lyapunov method and graph theory. Finally, an example showing the effectiveness of the provided criterion is given. [Copyright &y& Elsevier]

Abstract: Two new stability results, Hyers–Ulam–Rassias stability and Hyers–Ulam stability, of a certain fractional integral equation with delay are presented by using a fixed point theorem in a generalized complete metric space. Some applications of our results are also provided. [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: In the machining process, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece, hence the relevant measures must be taken to predict and avoid this phenomenon of instability. In this paper, we propose a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. It is proved that Hopf bifurcation exists when bifurcation parameter equals a critical value, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are given by using the normal form method and center manifold theorem. Numerical simulations show excellent agreement with the theoretical results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a commensalism model with a discrete delay and a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the interior equilibrium is investigated. It is found that the interior equilibrium is asymptotically stable when the discrete delay is less than a certain critical value and unstable when the discrete delay is greater than this critical value. By regarding the delay as the bifurcation parameter, the existence of Hopf bifurcation is also considered. Furthermore, the properties of Hopf bifurcation are determined. In particular, the global existence of bifurcated periodic solutions is established by applying the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, numerical simulations supporting the theoretical analysis are also included. [Copyright &y& Elsevier]