Time-delay Systems Research Articles

An effective numerical method for solving fractional delay differential equations using fractional-order Chelyshkov functions

In this paper, the fractional-order Chelyshkov functions (FCHFs) and Riemann-Liouville fractional integrals are utilized to find numerical solutions to fractional delay differential equations, by transforming the problem into a system of algebraic equations with unknown FCHFs coefficients. An error bound of FCHFs approximation is estimated and its convergence is also demonstrated. The effectiveness and accuracy of the presented method are established through several examples. The resulting solution is accurate and agrees with the exact solution, even if the exact solution is not a polynomial. Moreover, comparisons between the obtained numerical results and those recently reported in the literature are shown.

On hybrid dynamical systems of differential–difference equations

In this paper, we define and study a class of linear hybrid dynamical systems characterized by differential–difference equations. We introduce two operators that facilitate the analysis of these systems and derive explicit formulas for their solutions. We examine the transfer function matrix and characteristic polynomial to assess stability. Our theoretical findings are supported by numerical examples, demonstrating their application in power systems stability analysis. Specifically, we substantiate our theory within the context of power systems stability analysis, incorporating elements of discrete behavior.

Finite-time adaptive fuzzy event-triggered control for nonlinear time-delay system with input quantization via command filter

This paper investigates the problem of finite-time adaptive fuzzy event-triggered control for a class of uncertain nonlinear systems with input quantization. The nonlinear systems under consideration contain unmodeled dynamics and time-varying delays. Fuzzy logic systems are used to handle unknown nonlinear terms. Additionally, finite-time filters are introduced to solve the problem of “explosion of complexity” as well as to ensure that the derivative of the virtual controller can be approximated in finite time. The Lyapunov–Krasovskii function is used to handle time-varying delays. The hysteresis quantizer is used to ensure adequate precision when there is a low transmission rate requirement. The control strategy combines event-triggered mechanisms and quantization technology to ensure that all signals of the closed-loop system remain bounded in finite time. Finally, two examples are included to demonstrate the effectiveness of the proposed controller.

Differential Difference Equations in Trajectory Planning and Control for Differential Drive Robots: A Comprehensive Exploration

This paper provides a thorough investigation into the utilisation of differential difference equations (DDEs) for the purposes of trajectory planning and control in the context of differential drive four-wheeled robots. The inclusion of sensor and control delays is crucial when developing navigation strategies that are both resilient and efficient for robots functioning in dynamic environments. Differential delay equations (DDEs) provide a robust mathematical framework for representing the dynamics of such systems, facilitating precise and reliable path tracking. In order to demonstrate the versatility and practicality of Delay Differential Equations (DDEs), we investigate three distinct scenarios: straight-line path tracking, circular path tracking, and path planning with obstacle avoidance. These scenarios serve as demonstrations of how Delay Differential Equations (DDEs) facilitate the robot's ability to adjust its motion by utilising previous information, thereby ensuring a trajectory tracking process that is both smooth and stable. The trajectory plots provide a visual representation of the robot's path and effectively illustrate the successful completion of various navigation objectives. This study highlights the importance of dynamic differential equations (DDEs) in the advancement of intelligent and adaptable robotic systems, thereby paving the way for future progress in the realm of robotics and autonomous systems.

Modeling Population Dynamics in the Indian Context: A Differential Difference Equation Approach

The study of population dynamics is of paramount importance in comprehending the evolving demographics and socioeconomic structure of a country. The task of modelling population dynamics in the Indian context poses distinct challenges and opportunities owing to the extensive and heterogeneous nature of the country's population. This research paper utilises a differential difference equation (DDE) methodology to comprehensively model population dynamics in India. The study seeks to enhance comprehension of population dynamics by considering various factors, including birth rates, death rates, migration patterns, and seasonal variations. The study employs Delay Differential Equations (DDEs) as a mathematical framework to effectively model the dynamic characteristics of population systems. These equations account for time delays, historical events, and discrete factors that play a significant role in shaping population dynamics, whether it be growth or decline. Numerical techniques, such as Euler's method and the Runge-Kutta method, are utilised in order to solve Delay Differential Equation (DDE) models and effectively simulate the dynamics of populations over a given time period. The evaluation of each method involves a comparative analysis of its accuracy, smoothness, convergence properties, and computational efficiency. The research findings presented in this study enhance our comprehension of population dynamics in India and offer significant insights that can be utilised by policymakers, urban planners, and researchers in the fields of demography, public health, and social sciences. The results have the potential to provide valuable insights for evidence-based decision-making and the development of effective policies aimed at addressing the various challenges and opportunities related to population dynamics in the Indian context.

A Review of Differential Equations that is applicable to Differential Difference Equations

Mathematical modeling relies heavily on differential equations because they offer a robustframework for studying and understanding dynamical systems. However, the study of differential differenceequations (DDEs) has evolved as a subfield within differential equations due to the increasing prevalence ofsystems with discrete time steps and delayed effects. The purpose of this work is to present a synopsisof research on differential equations that may be used in the analysis of differential difference equations.The review starts with a thorough introduction to ODEs and PDEs (ordinary and partial differentialequations). Fundamental ideas, analytical approaches, and numerical strategies for solving thesecontinuous-time problems are discussed. ODEs and PDEs are highlighted for their importance insimulating a wide range of physical, biological, and engineering systems.

Dynamic impact analysis of the time-delay levitation control system on maglev vehicle system after adding smith predictor

The time delay (TD) in the levitation control system significantly affects the dynamic performance of the closed-loop system in electromagnetic suspension (EMS) maglev vehicles. Excessive TD can cause levitation instability, making it essential to explore effective mitigation methods. To address this issue, a Smith Predictor (SP) is integrated into the traditional PID levitation control system. The combination of theoretical analysis and numerical simulation is employed to assess the stability of the time-delay levitation control system after the integration of the Smith Predictor. Theoretical analysis reveals that when TD exceeds a critical threshold, the levitation system becomes unstable. The addition of SP alters the root trajectory of the system characteristic equation from positive to negative, and recovers the levitation system to stable status. Assuming complete knowledge of the dynamic system, the TD compensation value in the SP becomes a key parameter that determines its performance. A minimum effective value (MEV) for TD compensation is identified, correlating with the system's stability region. Under the influence of TD, more complex systems and higher running speeds of the maglev vehicle lead to a narrower stable region and a larger MEV for TD compensation. Given the simulation parameters in this paper, with a system TD of 15 ms and a maximum vehicle speed of 160 km/h, the MEV for TD compensation in the SP should be set at 12 ms.

Suppression and synchronization of chaos in uncertain time-delay physical system

Suppression and synchronization of chaos in uncertain time-delay physical system

Improved Mathematical Models of Parkinson's Disease with Hopf Bifurcation and Huntington's Disease with Chaos.

Using delay differential equations to study mathematical models of Parkinson's disease and Huntington's disease is important to show how important it is for synchronization between basal ganglia loops to work together. We used the delay circuit RLC (resistor, inductor, capacitor) model to show how the direct pathway and the indirect pathway in the basal ganglia excite and inhibit the motor cortex, respectively. A term has been added to the mathematical model without time delay in the case of the hyperdirect pathway. It is proposed to add a non-linear term to adjust the synchronization. We studied Hopf bifurcation conditions for the proposed models. The desynchronization of response times between the direct pathway and the indirect pathway leads to different symptoms of Parkinson's disease. Tremor appears when the response time in the indirect pathway increases at rest. The simulation confirmed that tremor occurs and the motor cortex is in an inhibited state. The direct pathway can increase the time delay in the dopaminergic pathway, which significantly increases the activity of the motor cortex. The hyperdirect pathway regulates the activity of the motor cortex. The simulation showed bradykinesia occurs when we switch from one movement to another that is less exciting for the motor cortex. A decrease of GABA in the striatum or delayed excitation of the substantia nigra from the subthalamus may be a major cause of Parkinson's disease. An increase in the response time delay in one of the pathways results in the chaotic movement characteristic of Huntington's disease.

A delay differential equation model on covid-19 with vaccination strategy

In this paper, we have extended SEIR model of COVID-19. The model incorporates two vital aspects in the form of vaccine compartment and constant time delay. The vaccination and time delay provide the information about immune protection and actual existence of the infection among the individuals, respectively. The model is analysed numerically and numerical simulation are executed for three different initial histories and constant time delays which affirm the biological relevance of the system. The analysis includes disease-free equilibrium (DFE), endemic equilibrium, and the basic reproduction number. The stability analysis is performed which reveal the asymptotic stability of the DFE when the basic reproduction number R0 < 1. The study addresses the boundedness and positivity of the solution as the time delay approaches zero. In addition, sensitivity analysis and contour plots for R0 with different parameters offer deeper insights into the model. The impact of vaccination and vaccine inefficacy on the model dynamics is explored.

Study of Caputo fractional derivative and Riemann–Liouville integral with different orders and its application in multi‐term differential equations

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 < 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 < 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality

Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality

Mathematical modeling of Ebola using delay differential equations

Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh–Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.

ASYMMETRIC FUNCTIONAL-BASED APPROACHTO EXPONENTIAL STABILITY OF LINEAR DISTRIBUTEDTIME-DELAY SYSTEMS

In this paper, the problem of exponential stability of linear time-delay systems with mixed discrete and distributed delays is studied. Based on an asymmetric Lyapunov-Krasovskii functional approach, sufficient conditions are derived in terms of linear matrix inequalities to guarantee the exponential convergence of the system state trajectories with a prescribed decay rate. The efficacy of the obtained results is demonstrated by a given numerical example and simulations.

Adaptive distributed fault estimation observer design for nonlinear time-delay multi-agent systems with directed graphs

This paper focuses the development of an adaptive observer-based distributed fault estimation observer (DFEO) for multi-agent nonlinear time-delay systems under a directed communication topology. The process involves constructing a fault estimation observer for each agent based on their relative output estimation errors. An adaptive DFEO design is proposed for multi-agent systems with a directed communication topology. Furthermore, investigating the effect of faulty agents on the criteria of fault detection on healthy agents. The proposed design is represented with Linear Matrix Inequalities (LMIs) to establish a systematic design procedure. The study also includes simulation results to demonstrate the effectiveness of the proposed design techniques.

Control and stochastic dynamic behavior of Fractional Gaussian noise-excited time-delayed inverted pendulum system

In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H∈(1/2,1), we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic Itoˆ equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system.

Adaptive Finite-Time-Based Neural Optimal Control of Time-Delayed Wheeled Mobile Robotics Systems.

For nonlinear systems with uncertain state time delays, an adaptive neural optimal tracking control method based on finite time is designed. With the help of the appropriate LKFs, the time-delay problem is handled. A novel nonquadratic Hamilton-Jacobi-Bellman (HJB) function is defined, where finite time is selected as the upper limit of integration. This function contains information on the state time delay, while also maintaining the basic information. To meet specific requirements, the integral reinforcement learning method is employed to solve the ideal HJB function. Then, a tracking controller is designed to ensure finite-time convergence and optimization of the controlled system. This involves the evaluation and execution of gradient descent updates of neural network weights based on a reinforcement learning architecture. The semi-global practical finite-time stability of the controlled system and the finite-time convergence of the tracking error are guaranteed.

Efficiency of Runge-Kutta-Fehlberg Method in Resolving Uncertainty: a Fuzzy Approach to Delay Differential Equations

Efficiency of Runge-Kutta-Fehlberg Method in Resolving Uncertainty: a Fuzzy Approach to Delay Differential Equations

On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots

On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots