Time-dependent Delay Research Articles

Application of fractional differential transform method and Bell polynomial for solving system of fractional delay differential equations

In this article, a new numerical technique is presented to obtain numerical solution of a system of fractional delay differential equations (FDDE’s) involving proportional and time dependent delay terms. The fractional derivative is used in Caputo sense. The proposed technique is the combination of fractional differential transform and Bell polynomial. The existence and uniqueness results are discussed for FDDE’s. Three numerical problems are discussed to show reliability and efficiency of the method. Numerical results are compared with exact and Matlab DDENSD solution. The main advantage of the present method is handing effectively the nonlinear terms present in the FDDEs by using Bell polynomial. The present method can deal with both linear and nonlinear FDDEs. The convergence result is discussed, and error analysis is presented in detail.

Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

Research in impulsive delay differential equations has been undergoing some exciting growth in recent times. This to a large extent can be attributed to the quest by mathematicians in particular and the science community as a whole to unveil nature the way it truly is. The realization that differential equations, in general, and indeed impulsive delay differential equations are very important models for describing the true state of several real-life processes/phenomena may have been the tunic. One can attest that in most human processes or natural phenomena, the present state is most often affected significantly by their past state and those that were thought of as continuous may indeed undergo abrupt change at several points or even be stochastic. In this study, a special strictly ascending continuous delay is constructed for a class of system of impulsive differential equations. It is demonstrated that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. The integral equivalence of the formulated system of equations is also obtained via a scheme similar to that of Perron by making use of certain assumptions.

Strong and weak divergence of the backward Euler method for neutral stochastic differential equations with time-dependent delay

. In this article, we consider the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay and reveal conditions of the divergence of the p-th absolute moments of the corresponding approximate solutions when p ∈ ( 0 , ∞ ) . This implies the strong L p -divergence of the method at finite time, for p ∈ [ 1 , + ∞ ) , and shows that its numerically weak convergence fails to hold. This article is motivated by the paper Hutzenthaler, M., Jentzen, A., Kloeden, P. E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A 467 (2011), no. 2130, 1563–1576, where a class of ordinary stochastic differential equations with superlinearly growing coefficients is studied.

Adaptive synchronization of the switching stochastic neural networks with time-dependent delays

Adaptive synchronization of the switching stochastic neural networks with time-dependent delays

Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

General stability of a triple layer beam with time-varying delay and weak internal damping

Analyzing a triple layer Rao–Nakra beam model which has time-dependent weight function and delay, weak internal damping, with thermal dissipation elements, is our focus. Heat conduction from Fourier’s law was adopted to regulate the thermal dissipation. It is known that when the three equations of the Rao–Nakra beam is directly and globally damped, an exponential stability is achieved. Weak internal damping leads to prolonged resonance, extended vibrational amplitudes and periods, as well as delayed energy dissipation. In this work we will rigorously demonstrate that despite weak internal damping and delay term in the bottom layer’s equation, we can obtain a general stability for the Rao–Nakra beam model. As a result, exponential and polynomial decay can be deduced as particular instances.

Synchronization of Stochastic Neural Networks Using Looped-Lyapunov Functional and Its Application to Secure Communication.

This study aims to investigate the synchronization of user-controlled and uncontrolled neural networks (NNs) that exhibit chaotic solutions. The idea behind focusing on synchronization problems is to design the user-desired NNs by emulating the dynamical properties of traditional NNs rather than redefining them. Besides, instead of conventional NNs, this study considers NNs with significant factors such as time-dependent delays and uncertainties in the neural coefficients. In addition, information transmission over transmission may experience stochastic disturbances and network transmission. These factors will result in a stochastic differential NN model. Analyzing the NNs without these factors may be incompatible during the implementation. Theoretically, the model with stochastic disturbances can be considered a stochastic differential model, and the stability conditions are derived by employing Itô's formula and appropriate integral inequalities. To achieve synchronization, the sampled-data-based control scheme is proposed because it is more effective while information is being transmitted over networks. In contrast to the existing studies, this study contributes in terms of handling stochastic disturbances, effects of time-varying delays, and uncertainties in the system parameters via looped-type Lyapunov functional. Besides this, in the application view, delayed NNs are employed as a cryptosystem that helps to secure the transmission between the sender and the receiver, which is explored by illustrating the statistical measures evaluated for the standard images. From the simulation results, the proposed control and derived sufficient conditions can provide better synchronization and the proposed delayed NNs give a better cryptosystem.

On the boundary stabilization of the KdV–KdV system with time-dependent delay

The boundary stabilization problem of the Boussinesq KdV–KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato’s approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV–KdV system decays exponentially by employing the Lyapunov method.

Switching attack modeling and vulnerability analysis of generator for dynamics study

This paper presents novel insights about cyber-attacks in the power network and their consequence in terms of stability. Compared to existing cyber-attack related studies in power systems, the work discusses switching attack construction and evaluates the impact on generator dynamics. As such, the study is in two folds: (i) switching attack construction, with selection of switching instants based on system states and (ii) vulnerability analysis of power system components, circuit breaker, governor system and excitation system of generators. The role of inherently present communication delay/state variable delay becomes important as it has direct impact on dynamic performance of the system. The impact of the constant and timedependent delay on the stability is addressed by determining the time-delay margin using Lyapunov-Krasovskii functional. Thereafter, the condition for successful switching attack construction is given in terms of delay margin. The accuracy of the derived condition for successful switching attack is demonstrated via real-time co-simulation (RTDS) and MATLAB software. The vulnerability of power system components in IEEE 3-machine, 9-bus network are analysed as individuals or their combinations when subjected to switching attacks. The adequacy related to power system stability as a consequence of cyber-attacks is investigated with an estimation of RoCoF of generators.

Strong Convergence and Asymptotic Exponential Stability of Modified Truncated EM Method for Neutral Stochastic Differential Equations with Time-dependent Delay

Strong Convergence and Asymptotic Exponential Stability of Modified Truncated EM Method for Neutral Stochastic Differential Equations with Time-dependent Delay

Stability results for the KdV equation with time-varying delay

In this paper, we consider the Korteweg–de Vries equation with time-dependent delay on the boundary or internal feedbacks. Under some assumptions on the time-dependent delay, on the weights of the feedbacks and on the length of the spatial domain, we prove the exponential stability results, using appropriate Lyapunov functionals. We finish by some numerical simulations that illustrate the stability results and the influence of the delay on the decay rate.

Physics-informed neural networks for integrated traffic state and queue profile estimation: A differentiable programming approach on layered computational graphs

This paper presents an integrated framework for physics-informed joint traffic state and queue profile estimation (JSQE) on freeway corridors, utilizing heterogeneous data sources. The integrated modeling framework aims to maximize the benefits of information from physics-informed analytical traffic flow models and field observations while leveraging joint estimation spaces. Potential inconsistencies between different modeling components must be acknowledged and carefully managed to ensure model feasibility. To minimize such inconsistencies, a nonlinear programming model is formulated for the JSQE problem, taking into account traffic flow models and observations from both corridor and local segment levels. At the corridor level, a fluid queue approximation is employed to model queuing dynamics. Assuming polynomial arrival and departure rates, critical system variables such as time-dependent delay, travel time, and queue length are analytically derived. To preserve the differentiability of traffic state variables, continuous space–time distribution functions are introduced to model traffic flow variables and partial differential equations. A computational graph is constructed to represent the nonlinear programming model in a layered structure, which is then solved using a forward–backward method. Comprehensive numerical experiments based on real-world and hypothetical datasets are designed to demonstrate the effectiveness of the proposed framework.

Stability of Cohen–Grossberg Neural Networks with Time-Dependent Delays

Stability of Cohen–Grossberg Neural Networks with Time-Dependent Delays

Uniform stability of a strong time-delayed viscoelastic system with Balakrishnan–Taylor damping

This paper studies a Balakrishnan–Taylor viscoelastic wave equation with strong time-dependent delay. Under suitable assumptions on the coefficients of the delay term, we establish a generalized stability result, which improve some earlier results in the literature.

A general model of demand and discounting.

A simulation using a two-commodity demand model, with time as the constraint, replicated the primary findings from delay discounting experiments and introduces explicit terms for time elasticity and cross-price substitution into the delay discounting paradigm. A two-commodity temporal demand equation based on Hursh and Silberberg (2008) and Hursh (2014) was used to emulate delay discounting experiments. The own-price and cross-price demand curves intersected and plotting those indifference points emulated the usual hyperbolic discount function for substitutes. Simulations examined delay discounting in relation to (a) time elasticity of demand, (b) substitution between the delayed and immediate alternatives, and (c) amplitude of demand for the delayed alternative. The simulated discount functions with substitutes were hyperbolic. The discount rate was a direct function of increasing time elasticity and substitutability of delayed alternative demand, shifting the function toward an exponential model. Amplitude of demand for the delayed alternative was inversely proportional to discount rate and supported a hyperboloid model with a power function of time (Killeen, 2015; Rachlin, 2006). The emulation of cross-commodity discounting involving drugs points to amplitude and persistence of time-dependent demand and cross-commodity substitution as primary factors. This report describes the first general model of time-dependent demand and delay discounting. The model implicates cross-commodity substitution as a potential factor in delay discounting. In the context of substance use disorder, the model underscores the importance of defining the properties of multicommodity demand (time elasticity, substitution, and amplitude) specific to the commodities and context. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Random attractors and invariant measures for 3D stochastic globally modified Navier-Stokes equations with time-dependent delay and coefficient

In this article, we investigate the dynamics of three-dimensional, non-autonomous, and globally modified Navier-Stokes equations with time-dependent delay and additive noise. Based on the well-posedness of solutions, we demonstrate that the solution operators of such equations generate a non-autonomous random dynamical system. We further prove the existence and uniqueness of random attractors as well as the periodicity and the upper semicontinuity for this random dynamical system. We also prove the existence of invariant measures supported on the random attractors. Moreover, we rigorously find that every limit of a sequence of invariant measures must be an invariant measure of the corresponding limiting equations as the noise intensity $ \varepsilon\rightarrow0 $ or the modification parameter $ N\rightarrow N_0\in (0, +\infty) $.

A multi-host schistosomiasis model with seasonality and time-dependent delays

Schistosomiasis is a parasitic disease caused by Schistosoma Japonicum that is transmitted by freshwater snails. Humans and some mammals are hosts of Schistosoma Japonicum. In this paper, we propose a multi-host (human-snail-bovine) schistosomiasis model with seasonality and temperature-dependent delays. We first introduce the basic reproduction number $ \mathcal R_0 $ for the whole system and the basic reproduction number $ \mathcal {\hat R}_0 $ for the system in the absence of bovines, respectively. Then we determine the extinction and persistence of the disease in terms of $ \mathcal R_0 $ and $ \mathcal {\hat R}_0 $. In the presence of susceptible humans, snails and bovines, we show that the disease persists if $ \mathcal R_0>1 $ and dies out if $ \mathcal R_0<1 $. As an application of our model, we study the transmission tendency of the schistosomiasis in Hubei province, China. Our numerical simulations indicate that the schistosomiasis incidences in both humans and bovines decline in an oscillatory way, but cannot disappear completely in the near future. A sensitivity analysis suggests that enhancing environmental disinfection, increasing treatment rate for both infected humans and bovines, replacing cattle with machine, and controlling the growth of snail population should be the effective measures for the prevention and control of schistosomiasis.

RBL-2H3 Mast Cell Receptor Dynamics in the Immunological Synapse.

The RBL-2H3 mast cell immunological synapse dynamics is often simulated with reaction-diffusion and Fokker-Planck equations. The equations focus on how the cell synapse captures receptors following an immune response, where the receptor capture at the immunological site appears to be a delayed process. This article investigates the physical nature and mathematics behind such time-dependent delays. Using signal processing methods, convolution and cross-correlation-type delay capture simulations give a -squared range of 22 to 60, in good agreement with experimental results. The cell polarization event is offered as a possible explanation for these capture delays, where polarizing rates measure how fast the cell polarization event occurs. In the case of RBL-2H3 mast cells, polarization appears to be associated with cytoskeletal rearrangement; thus, both cytoskeletal and diffusional components are considered. From these simulations, a maximum polarizing rate ranging from 0.0057 s-2 to 0.031 s-2 is obtained. These results indicate that RBL-2H3 mast cells possess both temporal and spatial memory, and cell polarization is possibly linked to a Turing-type pattern formation.

General stabilization of non-autonomous hybrid systems with delays and random noises via delayed feedback control

The feedback control, derived from delayed state observations, was triumphantly imported to H∞ and exponentially stabilize an unstable nonlinear hybrid system with time-invariant delay and random noise, which is adapt to autonomous systems. Spontaneously, given an unstable non-autonomous hybrid system with time-dependent delay and random noise, can one import a feedback control, derived from delayed state and mode observations, to stabilize it? Fortunately, this will be well done by the approach of multiple Lyapunov functionals. Surprisingly, it is revealed that the stabilization can be achieved not only in the forms of H∞ stability and exponential speed but also of polynomial speed and general speed. Furthermore, the interesting result with respect to polynomial growth at most is as well yielded here.

Almost Periodicity in Time-Dependent and State-Dependent Delay Differential Equations

We study the response of various linear and nonlinear differential equations to almost periodic forced oscillations. A special attention is given to differential equations with state-dependent and almost periodic type delays. To the best of our knowledge, there are no results in the literature that address this problem.