Multiple Time Delays Research Articles

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

Finite‐time lag bipartite synchronization of double‐layer networks with non‐linear coupling strength and random coupling delays via T‐S fuzzy logic theory

AbstractThis article discusses finite‐time lag bipartite (FETLB) synchronization of double‐layer networks with non‐linear coupling strength and multiple time delays. The cooperative and competitive interactions between nodes are considered based on signed graphs. To address the non‐linear couplings strength, the T‐S fuzzy logic theory is used. An intermittent control approach is introduced to achieve FETLB synchronization, effectively minimizing control costs. Moreover, by the Lyapunov functional method, we derive criteria for achieving FETLB synchronization and provide estimations for the synchronization settling time. In addition, numerical simulation affirms the validity of the theoretical findings, showcasing the practical application of the synchronization results in secure communication.

Event-based nonfragile state estimation for memristive neural networks with multiple time-delays and sensor saturations

This article investigates the issue of nonfragile state estimation (SE) for memristor-based neural networks (MNNs) with leakage delay and proportional delay. In actual engineering, a multitude of usefulness data are transmitted to the estimator through the networks, which stress the burden on communication bandwidth. A dynamic event-triggered mechanism (DETM) that relies on incomplete measurements is utilised to select valuable data. A novel delay-dependent criterion for the existence of the event-based state estimator is derived in terms of a convex optimisation problem by means of the Lyapunov theory and some integer inequalities technique. In the end, two numerical simulations are shown to illustrate the validity of the proposed theoretical methods.

A zonotope-based fault detection method for switched systems with parameter uncertainties and multiple time delays

This paper studies fault detection for discrete-time switched systems with parameter uncertainties and time delays. First, a state augmentation method is presented to decouple the effect of time delays. Then, a novel residual generator structure based on priori state estimation is introduced, and the adaptive thresholds for residual evaluation are computed via zonotope techniques. Meanwhile, peak-to-peak performance and H− performance are adopted to design the residual generator, ensuring both uncertainty robustness and fault sensitivity. Finally, numerical simulation results are provided to demonstrate the effectiveness and superiority of the proposed method.

Positivity and semi-global polynomial stability of high-order Cohen–Grossberg BAM neural networks with multiple proportional delays

In this paper, we study positivity and semi-global polynomial stability (PS) of higher-order Cohen-Grossberg BAM neural networks with multiple proportional time delays. The proportional delays considered here are unbounded and time-varying, differing from constant, bounded, and distributional time delays. The system model cannot be represented using vector and matrices, making certain approaches within the vector-matrix framework unsuitable for applying. To address this limitation, a direct method based on the solution of the system is proposed to provide sufficient conditions guaranteeing the positivity and semi-global polynomial stability (PS) of the model under consideration. Furthermore, the direct method is applied to establish global PS conditions for BAM neural networks with multiple proportional delays. The obtained conditions contain only a few simple linear scalar inequalities that are easily solved. The applicability of the obtained PS conditions is verified by two numerical examples, and the solution of a linear programming problem is also obtained based on these theoretical results. Notably, this method can be applied to many system models with proportional delays after minor modifications.

The impact of transcriptional and translational time delays on the cancer network regulated by MicroRNA‐17‐92

In this paper, we have formulated and analyzed a mathematical model with multiple time delays of the genetic regulatory network of the microRNA‐17‐92 (miR‐17‐92) cluster. We have focused on the effect of the associated transcriptional and translational time delays of both positive and negative feedback loops on the miR‐17‐92 regulatory network. To gain better insight, we have explored the role of individual time delay in controlling the transition of the G1/S cell cycle process. We pay particular attention to the concentration of E2F and Myc, which trigger the protein ON and OFF switches. To analyze the situation mathematically, we employ delay‐dependent stability analysis and bifurcation analysis using delay differential equations and a biological study of the time‐delayed random transitions between the ON and OFF states for protein synthesis. Based on our study, we can conclude that the translational time delay can halt the healthy transition of cells in the G1/S cell cycle phase as the small increase in can destroy bistability of the system. Simulations are carried out to numerically show the solution trajectories under the combined effect of delays. Finally, we summarize the simulation results and the impact of delays on the dual behavior of miR‐17‐92 as it can act as an oncogene or as a tumor suppressor gene.

Stability and Hopf Bifurcation of a Cytokine-Enhanced HIV Infection Model with Saturation Incidence and Three Delays

To discuss the influence of cytokine-enhanced rate and antibody delay on viral infection, in this paper, we propose a cytokine-enhanced HIV model with saturation incidence, antibody immune response and time delays. Multiple time delays, namely intracellular delay, virus replication delay and antibody delay, are included. First, we prove that the model is well-posed. Then, we obtain two basic reproductive numbers of the model. Next, the stability of the equilibria is investigated by using Lyapunov functions and LaSalle’s invariance principle. The results show that intracellular delay and virus replication delay do not impact on the stability of the three equilibria. However, if antibody immune response delay is positive, we determine some conditions for stability switches of the endemic equilibrium by using antibody immune response delay as a bifurcation parameter. It is concluded that with the increase of antibody immune response delay, the endemic equilibrium loses its stability and the system generates Hopf bifurcation. Finally, the numerical simulation results also show the correctness of the theoretical results.

PD distributed control and stability analysis for DC microgrids with multiple time delays

Distributed control is widely used for coordination of multiple distributed generations (DGs) in a microgrid (MG), but the information transmission in practice is inevitably suffered from time delays, which will deteriorate system performance and even cause system instability. To solve this problem, a proportional-differential (PD) distributed control method for DC MGs is proposed in this paper. Firstly, a small-signal system model considering multiple time delays including measurement delay and communication delay is established. The introduction of time-delayed differential terms results in the system being transformed into a neutral time-delayed system (NTDS). Then, the time delay margin (TDM) under different delayed scenarios is accurately determined by the frequency-domain stability analysis, and the relationship between the TDM and the differential gain is quantified. Compared with the conventional proportional (P-type) distributed control method, the TDM is enhanced effectively. In addition, an indicator is proposed to quantify the enhancement effect of introducing differential terms on TDM. Finally, the effectiveness of the proposed method and the accuracy of the stability analysis are verified by simulation and hardware-in-loop (HIL) tests.

Combined Impact of Multidelay Feedback Number and Interval: A Novel Mechanism for Controlling the Stability of Stochastic Duffing Systems

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay [Formula: see text] as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay [Formula: see text] of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields.

A new analytical method for designing centralised PI controllers for unstable systems using a direct synthesis approach

This paper introduces a new analytical technique using a direct synthesis strategy to develop centralised proportional–integral (PI) controllers for multivariable processes. The current method design controller is focused on attaining the desired closed-loop response for multi input multi output (MIMO) processes that involve multiple time delays. The conventional multivariable PI controller is obtained by approximating the ideal multivariable controller by the Maclaurin series expansion. This is accomplished by choosing the desired closed-loop response order as the system order plus two. Subsequent analysis investigates the proposed method’s efficacy in designing multivariable PI controllers. Servo and regulatory problems studies were conducted to find the effectiveness of the proposed method and compared it with other methods recently reported in the literature, as well as assessment of performance indices like integral absolute error (IAE) and integral square error (ISE). The proposed controller's robustness is assessed by plotting the inverse maximum singular value against frequency, both input and output multiplicative uncertainties.

Expanding the clinical application of OPM-MEG using an effective automatic suppression method for the dental brace metal artifact

Optically pumped magnetometer magnetoencephalography (OPM-MEG) holds significant promise for clinical functional brain imaging due to its superior spatiotemporal resolution. However, effectively suppressing metallic artifacts, particularly from devices such as orthodontic braces and vagal nerve stimulators remains a major challenge, hindering the wider clinical application of wearable OPM-MEG devices.A comprehensive analysis of metal artifact characteristics from time, frequency, and time–frequency perspectives was conducted for the first time using an OPM-MEG device in clinical medicine. This study focused on patients with metal orthodontics, examining the modulation of metal artifacts by breath and head movement, the incomplete regular sub-Gaussian distribution, and the high absolute power ratio in the 0.5–8 Hz band. The existing metal artifact suppression algorithms applied to SQUID-MEG, such as fast independent component analysis (FastICA), information maximization (Infomax), and algorithms for multiple unknown signal extraction (AMUSE), exhibit limited efficacy. Consequently, this study introduced the second-order blind identification (SOBI) algorithm, which utilized multiple time delays for the component separation of OPM-MEG measurement signals. We modified the time delays of the SOBI method to improve its efficacy in separating artifact components, particularly those in the ultralow frequency range. This approach employs the frequency-domain absolute power ratio, root mean square (RMS) value, and mutual information methods to automate the artifact component screening process.The effectiveness of this method was validated through simulation experiments involving four subjects in both resting and evoked experiments. In addition, the proposed method was also validated by the actual OPM-MEG evoked experiments of three subjects. Comparative analyses were conducted against the FastICA, Infomax, and AMUSE algorithms. Evaluation metrics included normalized mean square error, normalized delta band power error, RMS error, and signal-to-noise ratio, demonstrating that the proposed method provides optimal suppression of metal artifacts. This advancement holds promise for enhancing data quality and expanding the clinical applications of OPM-MEG.

On fractional ring neural networks with multiple time delays: Stability and Hopf bifurcation analysis

Stability and Hopf bifurcation analysis of neural networks have become popular topics in recent years. In this paper, a large-scale fractional ring neural network with multiple time delays is considered. The model comprises three rings that share a single node. The effects of time delay and fractional order on its stability and Hopf bifurcation have been investigated. Firstly, the characteristic equation is obtained by using the Laplace transform. Then, the time delay and fractional order are set as a bifurcation parameter respectively. Correspondingly, the local stability and Hopf bifurcation conditions are derived by analyzing the characteristic equations. Finally, two examples are given to support the theoretical analysis. In fact, this article provides an effective method for analyzing the stability and Hopf bifurcation of fractional ring neural networks.

Hyers–Ulam–Rassias stability of fractional delay differential equations with Caputo derivative

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature.

A comprehensive study of spatial spread and multiple time delay in an eco-epidemiological model with infected prey

This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations.

Stability analysis of fractional difference equations with delay.

Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order difference," can also have a long-time memory. Therefore, the fractional difference equations with delay are an appropriate model in a range of systems. Even so, there are not many detailed studies available related to the stability analysis of fractional order systems with delay. In this work, we derive the stability conditions for linear fractional difference equations with an arbitrary delay τ and even for systems with distributed delay. We carry out a detailed stability analysis for the cases of single delay with τ=1 and τ=2. The results are extended to nonlinear maps. The formalism can be easily extended to multiple time delays.

Adaptive event-triggered [formula omitted] consensus secondary control for islanded microgrids under multiple time delays

In this paper, an adaptive event-triggered H∞ consensus control scheme is proposed for the secondary layer of AC MGs. The controller design problem of topological communication network with external disturbance, multiple delay, and communication burden is considered. Therefore, using the Lyapunov-Krasovskii theory, we obtain sufficient conditions for the robust consensus convergence of multi-agents under event-triggered and multiple time delays. Because traditional event-triggered often use fixed thresholds, it is challenging to balance reducing the communication burden and maintaining excellent system performance. Therefore, we design an adaptive event-triggered method that adaptively adjusts the trigger threshold according to the consistency error. Finally, a simulation system is established. The proposed control strategy has been validated through simulation results.

Non-square internal model control for mode transition of hybrid electric vehicles with multiple time delays

Mode transitions of multi-mode hybrid electric vehicles need careful coordination among the engine torque, motor torque, and clutch transmitted torque. However, the three torques have different actuation delays which make it difficult to ensure mode transition performance. A non-square internal model controller (IMC) is proposed in this paper for mode transition during which the input number is greater than the output number and each input has a different actuation delay. At first, the Moore-Penrose generalized inverse matrix is introduced to solve the inverse of the non-square matrix and make the IMC applicable. Secondly, the all-pole method is adopted to approximate the three delay models. Based on these specialized techniques, the tracking controller and anti-disturbance controller of the IMC are derived. Experiment results verify the effectiveness of the proposed controller.

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.

Dissipativity of Stochastic Competitive Neural Networks with Multiple Time Delays

The (Q,R,S)-γ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Q,R,S)-\\gamma -$$\\end{document}dissipative of stochastic competitive neural networks (SCNNs) with leakage delays and discrete delay is studied. Firstly, Lyapunov–Krasovskii functional is constructed, which studies the relationship between various types of time delays. Secondly, by using integral inequality technique, the linear matrix inequality (LMI) criterion of (Q,R,S)-γ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Q,R,S)-\\gamma -$$\\end{document}dissipative in the mean square sense of SCNNs is obtained. Furthermore, the obtained LMI criterion is extended to the passivity in the mean square sense, stability in the mean square sense, (Q,R,S)-γ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Q,R,S)-\\gamma -$$\\end{document}dissipative and passivity. Finally, the effectiveness of the obtained results are verified by numerical simulation.

Multiple time delay induced Hopf bifurcation of a cortex - basal ganglia model for Parkinson’s Disease

Multiple time delay induced Hopf bifurcation of a cortex - basal ganglia model for Parkinson’s Disease