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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]

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]

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]

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]

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]

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: 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 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 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]

Abstract: In this paper, a delayed differential equation model that describes infection of thymus with HIV-1 is considered. We first investigate the existence and stability of the equilibria and then we study the effect of the time delay on the stability of the infected equilibrium. Criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Finally, by using a delay as a bifurcation parameter, the existence of Hopf bifurcation is investigated. Numerical simulations are presented to illustrate the analytical results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the uniform exponential stability of ordinary and delay dynamic equations. After revealing the equivalence between various types of uniform exponential stability definitions on time scales with bounded graininess, and demonstrating their relation when the graininess is arbitrary, we confine our attention to the uniform exponential stability of ordinary dynamic equations. We introduce and prove the Bohl–Perron criterion for delay dynamic equations: if for any bounded right-hand side, the solution of the delay dynamic equation with bounded coefficients and delays is bounded, then the trivial solution of the equation is uniformly exponentially stable. We also obtain some corollaries of this criterion. Based on these results, explicit exponential stability tests are derived for delay dynamic equations with nonnegative coefficients, which are illustrated with an example on a nonstandard time scale. [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]

Abstract: In this paper, we consider a Timoshenko system with a delay term in the feedback and prove a stability result. The beam is clamped at the endpoints and has, in addition to an internal damping, a feedback with a delay. Under an appropriate assumption on the weights of the two feedbacks, we prove the well-posedness of the system and establish an exponential decay result for the case of equal-speed wave propagation. [Copyright &y& Elsevier]

Abstract: Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. Discussing the change of the number of eigenvalues with zero real part, we locate the boundary of the stability region and finally determine the largest stability region of trivial solution. We investigate the existence of bifurcation phenomena of codimension one/two of the trivial equilibrium by considering the intersections of some parameter curves, which, in the aτ-half parameter plane, correspond to zero root or pure imaginary roots. In particular, the equivariant bifurcation is studied because of the equivariance of the system. We also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant Pitchfork–Hopf bifurcation points, Hopf–Hopf bifurcation points, and some higher codimension bifurcation points. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcation occurs. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the effect of the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we study Eq. for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when [Copyright &y& Elsevier]

Abstract: This paper is concerned with the stability properties of theoretical solution and numerical solution to nonlinear neutral differential equations with two delays in Banach space. Some sufficient conditions for the stability and asymptotic stability of the systems are given. The numerical stability of θ-methods when applied to the systems is also considered. A series of new stability results of θ-methods are obtained. [Copyright &y& Elsevier]

Abstract: We investigate the behavior of population models in a periodically varying environment. In this paper, we introduce the Fox surplus harvesting model with delay We obtain explicit conditions for existence of global solutions, existence of a positive periodic solution, oscillation of all solutions about positive equilibrium and global attractivity of this equilibrium. Numerical simulations illustrate the effectiveness of the conditions for oscillations and stability of the Fox model. It was also found that oscillations in the intrinsic rate r(t) and the environmental carrying capacity K(t) contributed quantitatively and qualitatively to the system’s dynamics in significant ways, and cannot be neglected in mathematical models of biological systems. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with the numerical solution to initial value problems of neutral delay differential equations (NDDEs). The sufficient conditions for the stability and asymptotic stability of the general linear methods for a class R α,β of nonlinear systems of NDDEs are derived. Numerical test is given that confirm the theoretical results in the end. [Copyright &y& Elsevier]

Abstract: This paper considers the stability problem of nontrivial solutions of delay differential equations with state-dependent impulses. By using the concepts of reduced system and quasi-stability, the stability problem of a nontrivial solution of the impulsive delay differential equation with state-dependent impulses is converted to that of the trivial solution of the impulsive delay differential equation with fixed impulses. Several stability criteria are then established utilizing the known results for systems with fixed impulses. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the nonlinear stability properties of a class of linear multistep methods when applied to initial value problems of stiff delay differential equations in Banach spaces. For the test problem classes D(α, λ ∗, β) and D(α, λ ∗, δ, β) in Banach spaces, a series of new stability results of the linear multistep methods are obtained. [Copyright &y& Elsevier]

This paper deals with H-stability of the linear θ-method with a general variable stepsize for the system of pantograph equations with two delay terms. A sufficient condition such that the system of pantograph equations is asymptotically stable is derived. Furthermore, when the linear θ-method with a general variable stepsize is applied to this system, it is shown that the linear θ-method is H-stable if and only if 1/2<θ⩽1. [Copyright &y& Elsevier]

This paper deals with H-stability of Runge–Kutta methods with variable stepsize for pantograph equations. It is shown that the Runge–Kutta methods with a regular matrix A are H-stable if and only if the modulus of stability function at infinity is less than 1. Furthermore, the same conclusion can be obtained if the Runge–Kutta methods are stiffly accurate. [Copyright &y& Elsevier]

In this paper we obtain the 3/2 stability theorems for the delay differential equation with impulsesx′(t)+λx(t)+F(t,x(·))=0, t&ges;0, t≠tk,x(tk+)−x(tk)=Ik(x(tk)), k=1,2… [Copyright &y& Elsevier]

This paper mainly analyzes the well-posedness, and the stability analysis for the global solution of neutral stochastic differential delay equations (NSDDEs) driven by Lévy noises. By using an integral lemma and a Lyapunov function approach, the existence and uniqueness theorem is proved. Then, by using the inequality technique and the stochastic analysis theory, the exponential stability in p th(p ≥ 2) moment of such equations is discussed. By using another integral lemma, and using the Baralat lemma as well as the stochastic analysis, the almost surely asymptotic stability is also studied. Finally, one example is given to check the effectiveness of the findings derived. [ABSTRACT FROM AUTHOR]

The emergence of rumors is subjective and aggressive which will bring disturbance and misleading to people's lives. This paper is dedicated to comprehensively investigating the influence of forced silence function, time delay and network topology on rumor propagation in online social networks. Considering the network supervision such as forced silence and network topology, a novel delayed SIR(susceptible-propagating-recovery) epidemic-like rumor propagation model in both homogeneous and heterogeneous networks is proposed. By using the mean-field theory, we analyze the propagation dynamics of the model in detail. Firstly, we demonstrate the boundedness of the solutions, then we obtain the basic reproduction number and the existence of the rumor-prevailing equilibrium points. Secondly, the forward and backward bifurcation as well as the local and global stabilities of the equilibrium points in both homogeneous and heterogeneous networks are theoretically analyzed and revealed. Finally, the possible impact is analyzed through numerical simulations, and the factors affecting rumor propagation are theoretically analyzed, which prove the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Hopf bifurcation of a delay differential equation modeling two-enterprise interaction mechanism is investigated. For the demands of computer simulation applications, the discrete-time two-enterprise interaction model is proposed by using the forward Euler scheme. By discussing the characteristic equation of linearized part, the local stability criterion of the positive fixed point is presented. Time lag is selected as the bifurcation parameter. Then it is demonstrated that variation of time lag may result in Hopf bifurcation. Furthermore, derivations for the directions and stability of Hopf bifurcation are also given. For the purpose of controlling Hopf bifurcation, a time-delayed feedback control scheme is proposed. Finally, we simulate the main results. [ABSTRACT FROM AUTHOR]