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We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds. [ABSTRACT FROM AUTHOR]

This paper is concerned with equivariant normal forms of semilinear functional differential equations (FDEs) in general Banach spaces. The analysis is based on the theory previously developed for autonomous delay differential equations and on the existence of invariant manifolds. We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

Machine tool vibrations impose severe limitations on industry. Recent progress in solving for the stability behavior of delay differential equations and in modeling milling operations with time delay differential equations has provided the potential to significantly reduce the aforementioned limitations. However, industry has yet to widely adopt the current academic knowledge due to the cost barriers in implementing this knowledge. Some of these cost prohibitive tasks include time-consuming experimental cutting tests used to calibrate model force parameters and experimental modal tests for every combination of tool, tool holder, tool length, spindle, and machine. This paper introduces an alternative approach whereby the vibration behavior of a milling tool during cutting is used to obtain the necessary model parameters for the common delay differential equation models of milling. • Method to estimate milling model parameters from experimental time series. • Estimated parameters used to predict stability chart for range of cutting conditions. • Stability predictions show close match with experimental results. • Extends the spectral element method to incorporate steady-state vibration. • Automated approach for instrumented milling that makes manual modal test unnecessary. [ABSTRACT FROM AUTHOR]

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation x ′ (t) + c (t) x (τ (t)) = 0 , t ≥ 0 , where c is locally integrable of any sign, τ (t) ≤ t is Lebesgue measurable, lim t → ∞ ⁡ τ (t) = ∞ , we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the 3/2-stability criterion to the case of measurable parameters, improving 1 + 1 / e to the sharp 3/2 constant. [ABSTRACT FROM AUTHOR]

We consider stage-structured population models of intra- and inter-specific competition at immature life stages. A prototype delay model is derived for a single species that experiences larval competition. Its solutions are bounded for any birth function. Other ways of modelling the birth rate can lead to nonlinear integral equations. In some situations the technique of reducing an age-structured model to a system of delay equations applies. In the case of immature competition the delay equations cannot always be written down explicitly because their right hand sides depend on the solutions of the nonlinear ordinary differential equations that arise when one solves the nonlinear age-structured equations that determine the maturation rates in terms of the birth rates. This situation arises in the case of competition between two strains or species. However, in our two-strain competition model, vital properties of those right hand sides can be indirectly inferred using monotone systems theory. [ABSTRACT FROM AUTHOR]

In this work we present a quantitative measurement-based method to describe the stability behaviour of a periodic dynamical system. In-process response for impulse during milling is analysed to provide operational stability prediction. The proposed method combines chatter detection techniques with the theory of time-periodic delay differential equations applied for general milling operations. Signal processing based on dynamic modal decomposition approach not only presents qualitative properties (like stability/instability) but also quantifies a measure of stability through the determination of the Floquet multipliers. The corresponding monodromy operator of the milling process is approximated from the measured system's response during machining operation. By changing the technological parameters, the variation of the modulus of the Floquet multipliers can be monitored. The stability limit can precisely be interpolated with the unitary multiplier; furthermore, the stability limit can be extrapolated while the manufacturing parameters remain in the chatter-free region. The presented approach is validated by numerical results and laboratory tests. • Measurement-based method to quantify stability of milling operation. • In-process impulse response is analysed to provide operational stability prediction. • It combines chatter detection techniques with the theory of time-periodic systems. • Extrapolation of stability limit opens ways for efficient cutting parameter optimization. • Results are supported by theoretical, numerical and experimental parts. [ABSTRACT FROM AUTHOR]

Abstract: It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. [Copyright &y& Elsevier]

Abstract: A reaction–diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel. [Copyright &y& Elsevier]

Abstract: An equation , , is considered, where and are linear operators in a Hilbert space , , is a non-linear operator, and is a constant. Under some assumptions on and sufficient conditions are given for the solution to exist globally, i.e., for all , to be globally bounded, and to tend to zero at a specified rate as . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equation. We prove that these traveling wave fronts are exponentially stable to perturbation in some exponentially weighted spaces, when the initial perturbation around the traveling wave fronts decays exponentially as , but the initial perturbation can be arbitrary large in other locations. The time decay rate is also obtained by weighted energy estimates. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the stability analysis of a delay difference system of the form , , where a, b and c are real numbers and k is a positive integer. We establish some exact conditions for the zero solution of the system to be asymptotically stable. [Copyright &y& Elsevier]

Abstract: Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized. [Copyright &y& Elsevier]

Abstract: In this paper, we study the behavior of solutions of second order delay differential equation where , , , are real numbers, τ is positive real number. A basic theorem on the behavior of solutions is established. As a consequence of this theorem, a stability criterion is obtained. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays. Our approach is based on the Crandall–Liggett approximation and the properties of semigroups. [Copyright &y& Elsevier]

Abstract: The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction–diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption. [Copyright &y& Elsevier]

Abstract: In this paper we stochastically perturb the delay Lotka–Volterra model into the stochastic delay differential equation (SDDE) The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially be divided into two categories: [(i)] If the delay Lotka–Volterra model already has some nice properties, e.g., nonexplosion, persistence, and asymptotic stability, then the SDDE will preserve these nice properties provided the noise is sufficiently small. [(ii)] When the delay Lotka–Volterra model does not have some desired properties, e.g., nonexplosion and boundedness, the noise might make the SDDE achieve these desired properties. [Copyright &y& Elsevier]

For a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function, we obtain the existence, stability and limiting profile of a phase-locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory. [Copyright &y& Elsevier]

In this paper we study stability and asymptotic behaviour of a real two-dimensional system x′(t)=A(t)x(t)+B(t)x(t−r)+h(t,x(t),x(t−r)), where r>0 is a constant delay, A, B and h being the matrix functions and the vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex-valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov–Krasovskii functional. [Copyright &y& Elsevier]

Time delays produced by dispersal are shown to stabilize Lotka–Volterra predator–prey models. The models are formulated as integrodifferential equations that describe local predator–prey dynamics and either intrapatch or interpatch dispersal. Dispersing individuals may (or may not) differ in the duration of their trips; these differences are captured via a distributed delay in the models. Our results include those of previous studies as special cases and show that the stabilizing effect continues to operate when the dispersal process is modeled more realistically. [Copyright &y& Elsevier]