Time-dependent Delay Research Articles

Universal Dichotomy for Dynamical Systems with Variable Delay.

We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.

Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.

Almost sure exponential stability of the θ-Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay when θ ∈ [0; 1 2

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

Stability and attractivity for Nicholson systems with time-dependent delays

Stability and attractivity for Nicholson systems with time-dependent delays

On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model

We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.

Estimating the minimum delay optimal cycle length based on a time-dependent delay formula

For fixed time traffic signal control, the well-known Webster’s formula is widely used to estimate the minimum delay optimal cycle length. However, this formula overestimates the cycle length for high degrees of saturation. In this paper, we propose two regression formulas for estimating the minimum delay optimal cycle length based on a time-dependent delay formula as used in the Canadian Capacity Guide and the Highway Capacity Manual (HCM). For this purpose, we develop a search algorithm to determine the minimum delay optimal cycle length required for the regression analysis. Numerical results show that the proposed formulas give a better estimation for the optimal cycle length at high intersection flow ratios compared to Webster’s formula.

Determining the time-dependent effective relaxation time of biofunctionalized magnetic nanoparticles conjugated with biotargets by using a high-Tc SQUID-based ac susceptometer for a magnetic immunoassay

In this study, we determined the time-dependent effective relaxation time (τeff) of biofunctionalized magnetic nanoparticles (BMNs) that were conjugated with biotargets for magnetic immunoassays. The BMNs were anti-alphafetoprotein (AFP)-coated onto dextran-coated magnetic nanoparticles to yield Fe3O4–anti-AFP. The biotarget was AFP. The time-dependent phase delay θ(t) of magnetization relative to the applied field strength was determined using a homemade high-Tc superconducting quantum interference device-based ac susceptometer. The detected θ(t) was then analyzed to derive τeff(t), the results of which showed that τeff(t) increased over time. Additionally, the change in τeff after the BMNs conjugated with the AFP (Δτeff) increased with the AFP concentration (ΦAFP). We attribute this to the additional magnetization generated when the BMNs conjugated with the AFP, which thereby reduced the Néel and Brownian relaxations and consequently improved the τeff. We established the relationship between ΔτAFP and ΦAFP and thus provide a foundation for assaying unknown quantities of AFP.

Dynamical behavior of HIV-1 epidemic model with time dependent delay

The delayed HIV-1 infection mathematical model with two delays is proposed. One of which represents the latent period between the time of contacting and entering of virions into the target cells while the second one stands for virus production period between the new virions to be produced within and released from the infected cells. The basic reproduction number R0is found for the proposed model and it is proved that the uninfected steady state is globally asymptotically stable if R0 1. And if R0> 1, then an infected steady state occurs which is proved to be locally as well as globally asymptotically stable. The formulae for R0shows that it is the decreasing function of both delays.

Differentiability of solutions with respect to the delay function in functional differential equations

In this paper we consider a class of functional differential equations with time-dependent delay. We show continuous differentiability of the solution with respect to the time delay function for each fixed time value assuming natural conditions on the delay function. As an application of the differentiability result, we give a numerical study to estimate the time delay function using the quasilinearization method.

Effect compensation of the presence of a time-dependent interior delay on the stabilization of the rotating disk-beam system

In this article, we investigate the compensation problem of the effect of the interior time-dependent delay on the stabilization of a rotating disk-beam system. The physical system consists of a flexible beam free at one end, and attached to the center of the rotating disk whose angular velocity is time-varying. Assuming that a time-dependent interior delay is present in the system, we introduce a dynamic boundary force control at the free end of the beam and a torque control on the disk. Then, we show the destabilizing effect of the delay is compensated. Specifically, it is shown that the presence of such proposed controls assures the exponential stability of the system, provided that some reasonable conditions on the angular velocity of the disk and delay are fulfilled. Numerical examples in the case of constant delay are also provided to highlight the stability result.

An explicit analytic approximation of solutions for a class of neutral stochastic differential equations with time-dependent delay based on Taylor expansion

This paper represents a contribution to the analysis of approximate methods for stochastic differential equations based on the application of Taylor expansion, under the Lipschitz and linear growth conditions. The Lp and almost sure convergence of the appropriate approximate solutions are considered for a class of neutral stochastic differential equations with time-dependent delay. Coefficients of the approximate equations, including the neutral term, are Taylor approximations of the coefficients of the initial equation up to the first derivatives. For p ≥ 2, the rate of the Lp-convergence of the sequence of approximate solutions to the exact solution is estimated as 2lp−12l, where l > 1 is an integer. The presence of the neutral term in the equation reflected to the rate of convergence.

The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments

In this paper we consider stochastic differential equations with piecewise constant arguments. These equations describe hybrid dynamical systems, that is, combinations of continuous and discrete systems. Our aim is to establish the mean square convergence of the Euler–Maruyama approximate solution under quite general conditions, that is, the global Lipschitz condition and the linear growth condition, which guarantee the existence and uniqueness of the true solution. Then we show that the initial equation is exponentially stable in mean square if and only if, for some sufficiently small step-size Δ, the Euler–Maruyama method is exponentially stable in mean square. The stability study does not involve the Lyapunov functions nor functionals. It should be pointed out that stochastic differential equations with piecewise constant arguments can be regarded as a class of stochastic differential equations with multiple time-dependent delays which can be found in the literature. However, in the convergence analysis of the Euler–Maruyama method of these equations, it is required that the delay functions satisfy Lipschitz continuity condition. In the present paper, the delay functions do not satisfy that condition, so it represents an extension of results from the papers Mao (2003, 2007), using the technique similar to that from the first paper.

Orientational dynamics of a ferronematic liquid crystal in a rotating magnetic field

The behavior of the orientational structure of a ferronematic in a rotating uniform magnetic field is investigated using the continual theory. The time-dependent system of equations describing the dynamics of the ferronematic is derived. The dependences of the angles of rotation of the director and of the magnetization of the ferronematic on the velocity of field rotation are determined for various values of the material parameters. Two regimes (synchronous and asynchronous) of rotation of the ferronematic structure are detected. In the synchronous regime, the director rotates with the frequency of the magnetic field and a constant phase delay. The asynchronous regime is characterized by a time-dependent phase delay. The dependence of the critical angular velocity of magnetic field rotation, which determines the boundary between the synchronous and asynchronous regimes, on the magnetic field strength is derived.

Analysis of a general stochastic non-autonomous logistic model with delays and Lévy jumps

In this paper, a general stochastic non-autonomous logistic model with delays and Lévy jumps is proposed and studied. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the model are established. The threshold between weak persistence and extinction is also obtained. Finally, numerical simulations are introduced to illustrate the theoretical analysis results. Our results illustrate that both persistence and extinction have close relationships with time-dependent delay and Lévy noise.

A neuromorphic implementation of multiple spike-timing synaptic plasticity rules for large-scale neural networks.

We present a neuromorphic implementation of multiple synaptic plasticity learning rules, which include both Spike Timing Dependent Plasticity (STDP) and Spike Timing Dependent Delay Plasticity (STDDP). We present a fully digital implementation as well as a mixed-signal implementation, both of which use a novel dynamic-assignment time-multiplexing approach and support up to 226 (64M) synaptic plasticity elements. Rather than implementing dedicated synapses for particular types of synaptic plasticity, we implemented a more generic synaptic plasticity adaptor array that is separate from the neurons in the neural network. Each adaptor performs synaptic plasticity according to the arrival times of the pre- and post-synaptic spikes assigned to it, and sends out a weighted or delayed pre-synaptic spike to the post-synaptic neuron in the neural network. This strategy provides great flexibility for building complex large-scale neural networks, as a neural network can be configured for multiple synaptic plasticity rules without changing its structure. We validate the proposed neuromorphic implementations with measurement results and illustrate that the circuits are capable of performing both STDP and STDDP. We argue that it is practical to scale the work presented here up to 236 (64G) synaptic adaptors on a current high-end FPGA platform.

Permanence of a Gilpin-Ayala predator-prey system with time-dependent delay

In this paper, a Gilpin-Ayala predator-prey model with time-dependent delay with m preys and $(n-m)$ predators is studied, which can be seen as a modification of the traditional Lotka-Volterra competition predator-prey system model. Two sets of sufficient conditions on the permanence of the system are obtained. One set is delay dependent, while the other set is delay independent.

Predictor Design for Connected Cruise Control Subject to Packet Loss

In this work connected cruise control with digital controllers is investigated for a vehicular string on a single lane. The longitudinal control of each vehicle is based on the information broadcast by other vehicles via wireless communication. We demonstrate that intermittencies and packet loss introduce time-dependent time delays in the control loop that may destabilize the system. We analyze the plant and string stability and construct predictors to attenuate the destabilizing effects of the imperfections in the communication.

Stabilization of a non-linear rotating flexible structure under the presence of time-dependent delay term in a dynamic boundary control

This paper is concerned with the feedback stabilization problem for a rotating rigid body-beam system via a torque control exerted on the rigid body and a time-dependent delayed dynamic boundary force control acting at the free end of the beam. Sufficient conditions on the angular velocity of the rigid body and the delay term are derived to assure the existence and uniqueness of solutions to the system. Furthermore, the energy method approach is employed to establish the exponential stability of the closed-loop system under stronger but reasonable assumptions.

The effects of time-dependent delays on global stability of stochastic Lotka–Volterra competitive model

In this paper, a stochastic Lotka–Volterra competitive model with time-dependent delays is investigated. Sufficient conditions for global asymptotic stability of the positive equilibrium are established. The obtained result demonstrates that time-dependent delays have important impacts on the global asymptotic stability of the positive equilibrium of the considered system.

Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay

This paper represents the continuation of the analysis from papers Milošević (2011) [10] and Milošević (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function.