Time-delay Differential Equation Research Articles

Vibration suppression of suspended cables with three-to-one internal resonances via time-delay feedback

Based on the time-delay feedback control, the vibration suppression of suspended cables with three-to-one internal resonances are investigated. Initially, the nonlinear differential equation of motion for a suspended cable under time-delay feedback control is considered, and a discrete model is derived using the Galerkin method. Subsequently, the method of multiple scales is employed to perturbatively solve the discrete time-delay differential equation, determining the modulation equations around the first primary resonance. Steady-state and periodic solutions of the modulation equations are detected numerically. Numerical results indicate that the internal resonance enhances the nonlinear dynamical complexity of the controlled suspended cable. It is observed that the time delay and control gain affect the controlled system: in particular, an increase in control gain leads to a reduction in response amplitude. By adjusting the time delay and control gain, the critical excitation can be altered, an aspect that could be very useful from a pratical point of view. This research sheds light on the intricate dynamics of suspended cable and provides a theoretical foundation for designing more effective control strategies in engineering applications.

A Rumor Propagation Model Considering Media Effect and Suspicion Mechanism under Public Emergencies

In this paper, we collect the basic information data of online rumors and highly topical public opinions. In the research of the propagation model of online public opinion rumors, we use the improved SCIR model to analyze the characteristics of online rumor propagation under the suspicion mechanism at different propagation stages, based on considering the flow of rumor propagation. We analyze the stability of the evolution of rumor propagation by using the time-delay differential equation under the punishment mechanism. In this paper, the evolution of heterogeneous views with different acceptance and exchange thresholds is studied, using the standard Deffuant model and the improved model under the influence of the media, to analyze the evolution process and characteristics of rumor opinions. Based on the above results, it is found that improving the recovery rate is better than reducing the deception rate, and increasing the eviction rate is better than improving the detection rate. When the time lag τ < 110, it indicates that the spread of rumors tends to be asymptotic and stable, and the punishment mechanism can reduce the propagation time and the maximum proportion of deceived people. The proportion of deceived people increases with the decrease in the exchange threshold, and the range of opinion clusters increases with the decline in acceptance.

Existence and Uniqueness of Almost Periodic Solutions to Time Delay Differential Equations

Existence and Uniqueness of Almost Periodic Solutions to Time Delay Differential Equations

MODELING STOCHASTIC ADVERSE EFFECTS OF CBN 2023 REDESIGNED NAIRA NOTES POLICY ON RURAL FARMERS IN NIGERIA

The recent Central Bank of Nigeria (CBN) 2023 redesigned naira notes is of good benefits to strengthen the economy of the country by checking counterfeiting and hoarding of large volume of banknotes by the public. Despite all the efforts made by the CBN for citizens to enjoy the benefits of this implementation, most rural farmers were faced with adverse effects of uncertainties in the production and marketing of their agricultural produce due to lack of redesigned new naira notes in circulation. The adverse effects of these uncertainties are modeled as Advanced Stochastic Time-Delay Differential Equation (ASTDDE). The modeled equation is solved using Extended Second Derivative Block Backward Differentiation Formulae Method (ESDBBDFM) without the use of interpolation techniques in the evaluations of the delay term and noise term. In comparing the numerical results of this method with other existing methods in literature, the newly developed mathematical expressions for the evaluations of the delay term and the noise term in solving ASDDEs with the discrete schemes of ESDBBDFM gives better results for step number than step numbers and 3 by producing Least Minimum Absolute Random Error (LMARE) in a Lower Computational Processing Unit Time (LCPUT) faster than other existing methods that applied interpolation techniques in evaluations of the delay term and the noise term.

Nonlinear chatter and reliability analysis of milling Ti-6Al-4V with slender ball-end milling cutter

Slender ball-end milling cutters with high length/diameter ratio can offer increased accessibility in the deep cavity machining of parts in the aviation industry. In this paper, a comprehensive nonlinear machining dynamics model for milling Ti-6Al-4V with the slender ball-end milling cutter is proposed to predict and understand on cutting stability, vibration characteristics and dynamic reliability. The tool tip dynamics model of the slender ball-end milling cutter is established including the structural nonlinearity of bearing joints, the contact flexibility of joint interfaces and the influence of tool geometry. Generated nonlinear dynamic cutting force is calculated considering the multiple regenerative effect, the nonlinearity of tool away from cutting and the run-out of tool. The effect of process damping is included in the nonlinear time-delayed differential equation of the machining dynamics model. Furthermore, the reliability evaluation method for the dynamic accuracy of milling Ti-6Al-4V workpieces is established with the application of Monte Carlo method and active learning Kriging model. Some milling experiments on Ti-6Al-4V workpiece are designed to validate the proposed model by comparing the predicted and measured results. The milling stability, vibration characteristics and dynamic accuracy reliability of milling Ti-6Al-4V with the slender ball-end milling cutter are discussed. The increase in the fractal dimension of joint interfaces can lead to the improvement of milling stability, and increased fractal roughness results in reduced milling stability. The reliability in the dynamic accuracy of milling Ti-6Al-4V workpiece is 99.26 %, and the distribution parameters in the fractal dimension of joint interfaces have the greatest influence on the reliability of dynamic accuracy.

Extended State-Dependent Differential Riccati Equation (ESDDRE) Controller Design for a Chemical Plug-Flow Reactor via Time-Delay Partial Differential Equation

Extended State-Dependent Differential Riccati Equation (ESDDRE) Controller Design for a Chemical Plug-Flow Reactor via Time-Delay Partial Differential Equation

Stability Analysis of Milling Based on the Barycentric Rational Interpolation Differential Quadrature Method

Chatter causes great damage to the machining process, and the selection of appropriate process parameters through chatter stability analysis is of great significance for achieving chatter-free machining. This article proposes a milling stability analysis method based on the barycentric rational interpolation differential quadrature method (DQM). The dynamics of the milling process considering the regeneration effect is first modelled as a time-delay differential equation (DDE). When adjacent pitch angles of the milling cutter are symmetric, the milling dynamic equation contains a single time delay. Otherwise, when adjacent pitch angles are asymmetric, the dynamic equation contains multiple time delays. The barycentric rational interpolation DQM is then used to approximate the differential and delay terms of the milling dynamics equation, and to construct a state transition matrix between adjacent milling periods. Finally, the chatter stability lobe diagram (SLD) is obtained based on the Floquet theory. According to the SLD, the appropriate spindle speed can be selected to obtain the maximum stable axial depth of cutting, thereby effectively improving the material removal rate. The accuracy and efficiency of the proposed method have been validated by two widely used milling models, and the results show that the proposed method has good accuracy and computational efficiency.

Existence and stability of solution for time-delayed nonlinear fractional differential equations

ABSTRACT The objective of this study is to analyze the existence and stability of solutions for a mathematical model of nonlinear systems of time-delayed fractional differential equations involving Atangana-Baleanu-Caputo ( ABC ) fractional derivatives with vaccination. The epidemiological states of the total population are classified as susceptible individuals S ( t ) , infected individuals I ( t ) , quarantined individuals Q ( t ) , recovered individuals R ( t ) , vaccinated individuals V ( t ) and insusceptible/protected persons P ( t ) . By considering coordinate transformation, we linearize our system and obtain a matrix coefficient for the proposed time-delayed fractional differential equation. We establish sufficient requirements and inequalities using the generalized Gronwall inequality and Krasnoselskii's fixed-point theorem to prove the existence of solution for our problem. Additionally, we construct sufficient requirements for stability analysis by employing the Laplace transform and matrix measure theory. We also investigate criteria for the local stability of the epidemic and disease-free equilibrium points. Finally, we provide a numerical illustration to validate the effectiveness of our results.

A study on inventory control strategies of fresh food supply chain considering advertising delay effect

PurposeThe existence of the advertising delay effect and its impact on supply chain operations have been demonstrated in the current study. Therefore, this study develops a timely inventory control strategy for the fresh produce supply chain to address the advertising delay effect in the fresh produce supply chain.Design/methodology/approachThis study proposes a game model based on the Nerlove-Arrow time delay differential equation and Pontryagin's maximum principle. Through comparative analyses of the optimal equilibrium strategies, the authors compare the optimal equilibrium strategies, product goodwill and optimal inventory trajectories for suppliers and retailers under secondary replenishment decisions and decentralized decisions.FindingsThe authors find that (1) Only when the sales cycle meets certain conditions can the overall profit of the supply chain under the secondary replenishment decision be greater than that under the decentralized decision. As the price markup coefficient increases, the total profit of the supply chain first increases and then decreases. (2) With the increase in the delay time, the replenishment quantity during the initial period gradually decreases. After the delay time elapses, the inventory depletion rate under secondary replenishment decisions is faster than that under decentralized decision-making. (3) Although there is a continuously increasing maximum value of product goodwill with the increase in delay time, it becomes difficult to achieve this value for longer delays.Practical implicationsThe authors’ findings provide a theoretical basis for supply chain members of fresh agricultural products to select replenishment and inventory control strategies when adopting different levels of delay in advertising marketing.Originality/valueFirstly, this paper explains the impact of advertising delay effect on fresh produce supply chain from a dynamic perspective, and secondly, it provides guidance on advertising formulation and inventory replenishment for fresh produce retailers under the influence of advertising delay effect.

Cluster Synchronization of Multiple Fractional-Order Recurrent Neural Networks With Time-Varying Delays.

This article focuses on the cluster synchronization of multiple fractional-order recurrent neural networks (FNNs) with time-varying delays. Sufficient criteria are deduced for realizing cluster synchronization of multiple FNNs via a pinning control by applying an extended Halanay inequality applicable for time-delayed fractional-order differential equations. Moreover, an adaptive control applicable for the synchronization of fractional-order systems with time-varying delays is proposed, under which sufficient criteria are derived for realizing cluster synchronization of multiple FNNs with time-varying delays. Finally, two examples are presented to illustrate the effectiveness of the theoretical results.

Stability prediction via parameter estimation from milling time series

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.

Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

ABSTRACT In this article, a sub-optimal controller based on Extended State-Dependent Differential Riccati Equation (ESDDRE) is suggested for a class of time-delay nonlinear Partial Differential Equation (PDE) systems. At first, an extended pseudo linearisation presentation for parameterisation form using State-Dependent Coefficients (SDC) is proposed. In this presentation, all the time-delay parts are placed in system matrices as well as in input vectors. By defining a cost function and a Hamiltonian related to the PDE systems, the sub-optimal control law regarded on the ESDDRE is obtained. The stability of the closed-loop system based on the ESDDRE control approach is ensured by using a proper Lyapunov function and also Poincaré inequality. Numerical simulation results for three time-delay nonlinear PDE systems illustrate the appropriate performance of the proposed control approach.

Passively mode-locked high-frequency dual-VCSEL system.

Two VCSELs placed facing each other with one biased chip while the second chip is unbiased is shown as a promising alternative to the popularly used conventional SESAM mode-locked VECSEL to generate mode-locked pulses. We propose a theoretical model using time-delay differential rate equations and numerically show that the proposed dual-laser configuration functions as a typical gain-absorber system. Parameter space defined by laser facet reflectivities and current are used to show general trends in the exhibited nonlinear dynamics and pulsed solutions.

Global stability analysis of a COVID-19 epidemic model with incubation delay

<abstract><p>In this paper, we propose, analyze and simulate a time delay differential equation to investigate the transmission and spread of Coronavirus disease (COVID-19). The basic reproduction number of the model is determined and qualitatively used to investigate the global stability of the model's steady states. We use numerical simulations to support the analytical results in the study. From the simulation results, we note that whenever the basic reproduction number is greater than unity, the model solutions will be associated with periodic oscillations for a considerable time scale from the start before attaining stability. This suggests that the inclusion of the time delay factor destabilizes the endemic equilibrium point leading to periodic solutions that arise due to Hopf bifurcations for a certain time frame.</p></abstract>

Nonlinear Langevin time-delay differential equations with generalized Caputo fractional derivatives

Nonlinear Langevin time-delay differential equations with generalized Caputo fractional derivatives

Computational solution of advanced stochastic time-delay differential equations using hybrid block extended Adams Moulton methods for customer’s satisfaction when using electronic payment systems in Nigerian banks

This study examined the implementation of Hybrid Block Extended Adams Moulton Methods (HBEAMM) for the computational solution of Advanced Stochastic Time-Delay Differential Equations (ASTDDEs) through the use of various electronic payment systems such as ATM, POS and internet banking offered by the Nigerian banks. This work attempts to proffer solution to customer challenges in using the ATM of most banks in Nigeria. The challenges are considered in this work as capable of resulting to advanced time-delay and volatility noise. It is therefore modeled as advanced stochastic time-delay differential equation (ASTDDE) and solved using Hybrid Extended Block Adams Moulton Methods (HEBAMMs). The discrete schemes of the proposed method (HBEAMM) were obtained by continuous formulations of multistep collocation method by matrix inversion approach. The necessary and sufficient conditions for convergence and stability of the method were analyzed and proved satisfactory. The discrete schemes obtained were applied in solving some advanced stochastic time-delay differential equations and the results obtained revealed the degree of customers’ satisfaction in the use of e-payment systems. The accuracy of the method is compared with other existing methods and presented graphically to prove its superiority.

A New Approach to Approximate Solutions of Single Time-Delayed Stochastic Integral Equations via Orthogonal Functions

This paper proposes a new numerical method for solving single time-delayed stochastic differential equations via orthogonal functions. The basic principles of the technique are presented. The new method is applied to approximate two kinds of stochastic differential equations with additive and multiplicative noise. Excellence computational burden is achieved along with a O(h2) convergence rate, which is better than former methods. Two examples are examined to illustrate the validity and efficiency of the new technique.

High efficiency and precision approach to milling stability prediction based on predictor-corrector linear multi-step method

Regenerative chatter is the most important factor affecting the stability of the milling process. It is core for suppressing chatter and improving production efficiency to accurately and efficiently identify the stable region of milling chatter. Therefore, according to the theory of predictor-corrector, three predictor–corrector methods (PCM) are, respectively, proposed for the milling stability region by applying the fourth-order Adams-Bashforth-Moulton formula, Simpson formula, and Hamming formula. Firstly, the regenerative chatter milling process is described as a second-order time-delay differential equation (DDE) with periodic coefficients. Thus, the forced vibration time can uniformly be discretized as a time node set. Secondly, the fourth-order Adams-Bashforth formula is used to predict the displacement at every time node, whereas the fourth-order Adams-Moulton formula can be employed to correct this predicted value. In addition, the fourth-order Simpson formula and Hamming formula can also correct the predicted value. Thus, a higher precision discrete prediction-correction expansion is constructed for the transformation of DDE into the state transition express. The Floquet theory can be depended on to present the judgment criterion of milling stability. Moreover, finally, under the same milling process parameters, comparisons of both the stability lobe curve and the local discrete error curve show that the PCM has a faster convergence rate than the 1st-SDM (first-order semi-discretization method) and 2nd-FDM (second-order full-discretization method). This shows that the PCM can obtain better computational accuracy under the same discrete number, whereas the PCM is significantly higher computational efficiency over 1st-SDM and 2nd-FDM. Meanwhile, considering the actual machining environment, helix angle effect and multiple modes effect of the tool are analyzed; experimental verification considering multiple modes with helix angle further indicates the applicability of the PCM.

Multiplicity–Induced–Dominancy property for second–order neutral differential equations with application in oscillation damping

This paper addresses the exponential stability of linear time-delay systems of neutral type. In general, it is quite a challenge to establish conditions on the parameters of the system in order to guarantee such a stability. Recent works emphasized the link between maximal multiplicity and dominant roots. Indeed, conditions for a given multiple root to be necessarily dominant are investigated, this property is known as Multiplicity-Induced-Dominancy (MID). The aim of this paper is to explore the effect of multiple roots with admissible multiplicities exhibiting, under appropriate conditions, the validity of the MID property for second-order neutral time-delay differential equations with a single delay. The ensuing control methodology is summarized in a five-steps algorithm that can be exploited in the design of higher-order systems. The main ingredient of the proposed method is the dominancy proof for multiple spectral values based on frequency bounds established via integral equations. As an illustration, the stabilization of the classical oscillator benefits from the obtained results.

Optimal disturbances for periodic solutions of time-delay differential equations

Abstract A concept of optimal disturbances of periodic solutions for a system of time-delay differential equations is defined. An algorithm for computing the optimal disturbances is proposed and justified. This algorithm is tested on the known system of four nonlinear time-delay differential equations modelling the dynamics of the experimental infection caused by the lymphocytic choriomeningitis virus. The results of numerical experiments are discussed.