Discrete Time Delay Research Articles

New design of robust H∞ controllers for descriptor discrete time-varying delay equations with bounded disturbances

In this paper, a linear matrix inequality (LMI)-based design of robust [Formula: see text] controllers for linear descriptor discrete-time delay equations is proposed. The existing works of FTS and [Formula: see text] control are extended to the linear descriptor equations subjected to time-varying delays and bounded disturbances. First, sufficient LMI-based conditions are given to ensure the admissibility of such system, and then the conditions are applied to design [Formula: see text] controllers. A numerical example with simulations is given to illustrate the proposed results.

Guaranteed Cost Control of Impulsive Nonlinear Stochastic Systems With Mixed Time Delays

ABSTRACTThis article mainly addresses the guaranteed cost control (GCC) problem for a class of impulsive nonlinear stochastic systems with mixed time delays, where both distributed time delay as well as discrete time delay are considered. Utilizing the Hamilton–Jacobi inequalities technique, Lyapunov–Krasovskii functional (LKF) and stochastic control theory, a new delay‐dependent sufficient condition is acquired to ensure the asymptotic stability in probability (ASP) of the closed‐loop system (CLS). Moreover, the guaranteed cost controller is designed and the upper bound of cost function is given. At the same time, three corollaries are presented for the special circumstance without impulse, distributed time delays or time‐varying time delay, respectively. Finally, the feasibility of the theoretical results is verified by an example.

A dynamic analysis of a tourism-based socioecological system

The study proposes a dynamic analysis regarding interactions between the resources provided by the forest, the wildlife present inside or in the proximity of that environment and visitors revolving around the before mentioned socioecological framework. The mathematical model is described by a nonlinear system with three differential equations. The discrete time delay is introduced to illustrate the entire past impact of tourists on forest resources and wildlife. The basic assumption is that the wildlife species which inhabit the area are relying entirely on forest resources to meet their needs for food, shelter, and to attract tourists. Also, there is a positive correlation between ecotourism activities and the presence of forest resources and wildlife. The equilibrium states are determined, and they are subjected to a stability and bifurcation analysis. The study employs a Hopf bifurcation analysis in the neighborhood of the equilibrium states by choosing the time delay as the bifurcation parameter. The critical values of the time-delay that lead to oscillatory behavior are determined. Numerical simulations are carried out to show the system’s qualitative behavior in the vicinity of the equilibria.

Hopf bifurcation and stability analysis for a delayed equation with φ-Laplacian

A formal framework for the analysis of Hopf bifurcations for a kind of delayed equation with $\varphi$-Laplacian and with a discrete time delay is presented, thus generalizing known results for the sunflower equation given by Somolinos in 1978. Also, under appropriate assumptions we prove the gradient-like behavior of the equation which, in turn, implies the non-existence of nonconstant periodic solutions. Our conditions improve previous results known in the literature for the standard case $\varphi(x)=x$.

Spatio-temporal dynamics in a delayed prey–predator model with nonlinear prey refuge and harvesting

In the current study, we introduce a model for the temporal and spatial interactions between prey and predator. The model incorporates a nonlinear refuge mechanism for prey, along with linear harvesting of prey and nonlinear harvesting for predators. Initially, we examined the well-posed nature of the model by analyzing the presence of all feasible equilibria and investigating the corresponding dynamics. Following that, we delve into the dynamics of the temporal model, focusing specifically on aspects such as uniform boundedness, permanence, and stability of viable equilibria. We demonstrate analytically that the proposed model experiences transcritical, saddle–node, Hopf, and Bogdanov–Takens bifurcations. It shows a variety of intricate dynamics involving Generalized Hopf and double Hopf. Then, discrete-time delay effects arising from the gestation of predator species have been incorporated into the temporal system. Hopf bifurcation for the delay parameter was detected in this investigation. Subsequently, we established conditions for self-diffusion instability and Turing instability in the spatiotemporal model, both with and without delay, employing the homogeneous Neumann boundary condition. Moreover, the discussion of sensitivity analysis (PRCC) serves to illustrate how crucial parameters influence the dynamics of the system. In addition, we conduct numerical simulations aiming to corroborate and validate the analytical results obtained.

Coupled Lamé System with a Viscoelastic Term and a Strong Discrete Time Delays

Coupled Lamé System with a Viscoelastic Term and a Strong Discrete Time Delays

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

Existence of Traveling Waves of a Diffusive Susceptible–Infected–Symptomatic–Recovered Epidemic Model with Temporal Delay

The aim of this article is to investigate the existence of traveling waves of a diffusive model that represents the transmission of a virus in a determined population composed of the following populations: susceptible (S), infected (I), asymptomatic (A), and recovered (R). An analytical study is performed, where the existence of solutions of traveling waves in a bounded domain is demonstrated. We use the upper and lower coupled solutions method to achieve this aim. The existence and local asymptotic stability of the endemic (Ee) and disease-free (E0) equilibrium states are also determined. The constructed model includes a discrete-time delay that is related to the incubation stage of a virus. We find the crucial basic reproduction number R0, which determines the local stability of the steady states. We perform numerical simulations of the model in order to provide additional support to the theoretical results and observe the traveling waves. The model can be used to study the dynamics of SARS-CoV-2 and other viruses where the disease evolution has a similar behavior.

A study on the dynamics of a breast cancer model with discrete-time delay

This study aims to discuss the impact of discrete-time delay on the anti-tumor immune response against tumor growth, excess levels of estrogen, and the source rate of immune cells in a breast cancer model. The non-negativity and boundedness of the solutions of the model are discussed. The existence of equilibria and their stability are examined. It is found that if the estrogen level is normal and the source rate of immune cells is low, the stability of the model around the co-existing equilibrium switches to instability via a Hopf bifurcation as the time delay increases. To validate the theoretical findings, a few numerical examples have been presented. The main result of this study is that the growth of tumors can be controlled if the immune system quickly generates an anti-tumor immune response. However, if the immune system takes a longer time to generate anti-tumor immune responses, the tumor growth cannot be controlled, and the system becomes unstable, which may result in the further spread of the disease.

Research on the ellipsoidal boundary of reachable sets of neutral systems with bounded disturbances and discrete time delays

<abstract><p>This research focuses on the challenge of defining the ellipsoidal boundaries of the reachable set (RS) for neutral-type dynamical systems with time delays. A novel analytical approach is proposed, leveraging the development of new Lyapunov functions and matrix inequality techniques. These methods provide powerful tools for determining the ellipsoidal boundaries of the system's RS. A comparative analysis, supported by numerical examples, demonstrates that the approach outlined in this study can accurately identify smaller yet effective RS boundaries compared to existing literature. This precise boundary determination offers significant theoretical support for state estimation and control design in dynamical systems, thereby enhancing their effectiveness and reliability in real-world applications.</p></abstract>

Regulator Problem for Linear Discrete-Time Delay Systems with Symmetrical Constraints

Regulator Problem for Linear Discrete-Time Delay Systems with Symmetrical Constraints

A mathematical model for tumor-immune competitive system with multiple time delays

A mathematical model is proposed to study the complicated dynamics of tumor-immune interactions with three discrete time delays. The proposed model consists of nine coupled ordinary differential equations (ODEs) whose components are tumor cells, tumor-specific CD8+T cells, dendritic cells, macrophages, regulatory T-cells (Tregs), interleukin-10 (IL-10), IL-12, transforming growth factor-β (TGF-β) and interferon-γ (IFN-γ). By utilizing the quasi-steady-state approximation for cytokine concentrations, we obtain a system of four coupled ODEs. We introduce three discrete time delays into our deterministic system to better understand the tumor-immune dynamics. Basic properties of the system, including existence, uniqueness, positivity, boundedness, and uniform persistence are discussed. Stability analysis of both the delayed and non-delayed system has been performed and the conditions for stability and direction of Hopf bifurcation have been determined. Furthermore, we assess the length of time delay required to maintain the stability of period-1 limit cycle. Parameter estimation techniques are discussed and numerical simulations are provided to support our theoretical analysis. Notably, we observe that time delays do not significantly influence the system behavior for the existing set of parameters value.

Analysis of a delayed mathematical model for tumor-immune cell interactions with Holling type II functional response

In this study, we analyzed a three-dimensional nonlinear differential system considering Holling type II functional response that describes the dynamics of tumor cells, cytotoxic T lymphocytes, and helper T cells, with a single interaction delay. The linear stability of the internal equilibrium point and the presence of the Hopf bifurcation are examined, with the discrete time delay serving as the bifurcation parameter. To demonstrate the rich dynamic behavior of the model, we present numerical simulations with various values of the time delay τ and the attack rate of cytotoxic T lymphocytes on tumor cells (α1). These simulations exhibit the presence of periodic oscillations and tumor death with survival of the mentioned immune cells, for all α1 greater than a fixed threshold with or without delay.

Fixed/Preassigned-Time Synchronization of Fully Quaternion-Valued Cohen–Grossberg Neural Networks with Generalized Time Delay

This article is concerned with fixed-time synchronization and preassigned-time synchronization of Cohen–Grossberg quaternion-valued neural networks with discontinuous activation functions and generalized time-varying delays. Firstly, a dynamic model of Cohen–Grossberg neural networks is introduced in the quaternion field, where the time delay successfully integrates discrete-time delay and proportional delay. Secondly, two types of discontinuous controllers employing the quaternion-valued signum function are designed. Without utilizing the conventional separation technique, by developing a direct analytical approach and using the theory of non-smooth analysis, several adequate criteria are derived to achieve fixed-time synchronization of Cohen–Grossberg neural networks and some more precise convergence times are estimated. To cater to practical requirements, preassigned-time synchronization is also addressed, which shows that the drive-slave networks reach synchronization within a specified time. Finally, two numerical simulations are presented to validate the effectiveness of the designed controllers and criteria.

Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays

In this paper, we investigate two within-host mathematical models that are based on differential equations. These mathematical models include healthy cells, tachyzoites, and bradyzoites. The first model is based on ordinary differential equations and the second one includes a discrete time delay. We found the models’ steady states and computed the basic reproduction number R0. Two equilibrium points exist in both models: the first is the disease-free equilibrium point and the second one is the endemic equilibrium point. We found that the initial quantity of uninfected cells has an impact on the basic reproduction number R0. This threshold parameter also depends on the contact rate between tachyzoites and uninfected cells, the contact rate between encysted bradyzoite and the uninfected cells, the conversion rate from tachyzoites to bradyzoites, and the death rate of the bradyzoites- and tachyzoites-infected cells. We investigated the local and global stability of the two equilibrium points for the within-host models that are based on differential equations. We perform numerical simulations to validate our analytical findings. We also demonstrated that the disease-free equilibrium point cannot lose stability regardless of the value of the time delay. The numerical simulations corroborated our analytical results.

New Criteria for Analyzing the Permanence, Periodic Solution, and Global Attractiveness of the Competition and Cooperation Model of Two Enterprises with Feedback Controls and Delays

This paper studies a class of the non-autonomous competition and cooperation model of two enterprises involving discrete time delays and feedback controls. The paper proposes new criteria for analyzing the permanence, periodic solution, and global attractiveness of the model. The common mathematical techniques of the Lyapunov method, the continuation theorem, and the comparison principle are used in this paper. By means of the comparison principle and inequality techniques, the concept of permanence is investigated, which refers to the long-term survival of the enterprises within the competitive and cooperative framework. Meanwhile, using the continuation theorem, we establish conditions under which the system exhibits periodic behavior. Additionally, the global attractiveness of the system is derived by constructing multiple Lyapunov functionals. Finally, an example is presented to illustrate the applicability and validity of the proposed criteria in this paper. This example serves as a demonstration that showcases the main results derived from the analysis.

Dynamical analysis of a glucose-insulin regulatory system with insulin-degrading enzyme and multiple delays.

This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model.

Robust Tracking and Disturbance Rejection for Decentralized Neutral Distributed-time-delay Systems

An important problem in control engineering is to design a controller to achieve robust asymptotic tracking of certain reference signals despite certain disturbance inputs. In the present work, this problem, which is known as the robust servomechanism problem, is considered for decentralized linear time-invariant (LTI) neutral systems with distributed time-delay. However, the system is also allowed to have discrete time-delays besides distributed time-delays. The reference signals and the disturbance input are assumed to satisfy a LTI neutral delay-differential equation with distributed and/or discrete time-delays. The necessary and sufficient conditions for the existence of a controller which solves this problem are derived. The structure of this controller (when it exists) is also presented.

An analytical approach of applying the Lyapunov direct method to polynomial differential systems with discrete time delays

The Lyapunov direct method is a common and effective tool for discussing the global stability of dynamical systems. In this paper, we propose an analytical approach to apply this method for the global stability of the positive equilibrium of a polynomial differential system with discrete time delays, which includes how to construct an applicable Lyapunov functional and to verify the negative (semi-)definiteness of its derivative. Here the linear combination of the Volterra-type functions/functionals and their integrals plays an important role. Moreover, when the delayed system also has a boundary equilibrium, we show that the Lyapunov functional for the global stability of the positive equilibrium can be reformulated to establish the global stability of the boundary equilibrium. As an illustration, we apply this approach to a virus infection model with two delays.

Effects of the random walk and the maturation period in a diffusive predator–prey system with two discrete delays

This study aims to present a complete Hopf bifurcation analysis of a model describing the relationship between prey and predator. A ratio-dependent reaction–diffusion system with two discrete time delays operating under Neumann boundary conditions governs the model that represents this competition. The bifurcation parameter for the analysis is a delay parameter that reflects the amount of time needed for the predator to be able to hunt. Bilazeroğlu and Merdan’s algorithm (Bilazeroğlu et al., 2021), which is developed by using the center manifold theorem and normal form theory, is used to establish the existence of Hopf bifurcations and also the stability of the bifurcating periodic solutions. The same procedure is used to illustrate some specific bifurcation properties, such as direction, stability, and period. Furthermore, by examining a model with constant coefficients, we also analyze how diffusion and the amount of time needed for prey to mature impact the model’s dynamics. To support the obtained analytical results, we also run some numerical simulations. The results indicate that the dynamic of the mathematical model is significantly influenced by diffusion, the amount of time needed for the predator to gain the capacity to hunt, and the amount of time required for prey to reach maturity that the predator can hunt.