Discrete Delays Research Articles

Verified Integration of Differential Equations with Discrete Delay

Many dynamic system models in population dynamics, physics and control involve temporally delayed state information in such a way that the evolution of future state trajectories depends not only on the current state as the initial condition but also on some previous state. In technical systems, such phenomena result, for example, from mass transport of incompressible fluids through finitely long pipelines, the transport of combustible material such as coal in power plants via conveyor belts, or information processing delays. Under the assumption of continuous dynamics, the corresponding delays can be treated either as constant and fixed, as uncertain but bounded and fixed, or even as state-dependent. In this paper, we restrict the discussion to the first two classes and provide suggestions on how interval-based verified approaches to solving ordinary differential equations can be extended to encompass such delay differential equations. Three close-to-life examples illustrate the theory.

Stochastic HIV/AIDS dynamics with discrete and distributed delays

Stochastic HIV/AIDS dynamics with discrete and distributed delays

α− dissipative filtering for Singular Markov jump system with time delays

This paper considers the [Formula: see text] dissipative filtering problem for a class of Singular Markov jump systems (SMJSs) with distributed time delays and discrete time delays. First, using Lyapunov’s stability theory and combining delay partitioning technique, integral partitioning technique, and free weight matrix method, the sufficient conditions for stochastic admissibility and [Formula: see text] dissipation of system are studied. Then, a filtering design method based on linear matrix inequalities (LMIs) is given to make the filtering error system stochastically admissible and [Formula: see text] dissipative. Finally, numerical simulations verify the effectiveness of the resulting method.

Improved quasi‐uniform stability criterion of fractional‐order neural networks with discrete and distributed delays

AbstractThis article mainly investigates the quasi‐uniform stability of fractional‐order neural networks with time discrete and distributed delays (FONNDDDs). First, a novel fractional‐order Gronwall inequality with discrete and distributed delays (FOGIDDDs) is established; it can be used to study the stability of a variety of fractional‐order systems with discrete and distributed delays (FOSDDDs). Second, on the basis of this inequality and Leray‐Schauder alternative theorem, the existence and uniqueness results for the FONNDDDs are proved. Third, an improved criterion for the quasi‐uniform stability of FONNDDDs is obtained in terms of this inequality. Ultimately, one numerical example is provided to expound the effectiveness and the superiority of the proposed result.

Stability and Hopf Bifurcation Analysis of a Continuous Neural Network With Mixed Delays

A three-neuron network model with mixed delays involving multiple discrete and distributed delays is considered in this paper. Taking the discrete and distribute delays as bifurcation parameters respectively, we investigate the stability of the system structure and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the root of the characteristic equation of the linearized system at the equilibrium state of the nonlinear system. The intervals of parameters that make the system stable and unstable are also given. In addition, when the conditions of Hopf bifurcation theorem are satisfied, the calculation formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution are presented by means of the central manifold theorem and normal form method. Finally, numerical experiments are carried out to support the correctness of the theoretical results in different cases. It can be concluded that the oscillation and instability caused by delays obviously affect the stability of the network.

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

<p style='text-indent:20px;'>Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.</p>

TDS-CONTROL: a MATLAB package for the analysis and controller-design of time-delay systems*

This paper presents TDS-CONTROL, a MATLAB package for the analysis and design of controllers for linear time-invariant systems with discrete delays. The code is based on a state-space representation of such systems in terms of delay-differential algebraic equations with input, output, and state delays. As such, a broad class of (interconnected) systems can be considered, including neutral systems. The controller design algorithms are based on optimizing a certain objective function, such as the spectral abscissa, the H-infinity norm, or the pseudo-spectral abscissa, with respect to the controller parameters. For example, to design a stabilizing controller the spectral abscissa, i.e., the real part of the right-most characteristic root, is minimized. As a strictly negative spectral abscissa is a necessary and sufficient condition for stability, the presented design method is not conservative and a stabilizing controller can be computed whenever it exits. This comes however at the cost of having to solve a non-smooth, non-convex optimization problem. The class of considered controllers consists of static and dynamic output-feedback controllers. By allowing to fix certain entries in the controller matrices, it is also possible to design structured controllers such as decentralized, overlapping and PID controllers. Finally, as a wide-range of delay systems can be considered, the software package takes the sensitivity of certain quantities, such as the spectral abscissa and the H-infinity norm, with respect to infinitesimal delay perturbations explicitly into account.

Mathematical analysis of HIV infection of CD4+ T-cells with discrete delays

In this study, we introduce a discrete time to the model to describe the time delays between infection of a CD4+ T-cells, and the emission of viral particles on a cellular level. We begin by determining the existence and stability of the equilibrium. Further We investigate the global stability of the infection-free equilibrium and give sufficient condition for the local stability of the infected steady state is asymptotically stable for all delays. Finally, the numerical simulations are presented to illustrate the analytical results.

Dynamics of a delayed Lotka-Volterra model with two predators competing for one prey

<p style='text-indent:20px;'>In this paper, we study the local dynamics of a class of 3-dimensional Lotka-Volterra systems with a discrete delay. This system describes two predators competing for one prey. Firstly, linear stability and Hopf bifurcation are investigated. Then some regions of attraction for the positive steady state are obtained by means of Liapunov functional in a restricted region. Finally, sufficient and necessary conditions for the principle of competitive exclusion are obtained.</p>

Numerical Methods for Delay Differential Equations with Threshold State-Dependent Delay

Threshold delays arise naturally in a wide variety of dynamical systems. Standard routines for performing numerical analysis of delay differential equations (DDEs) typically handle discrete constant or state-dependent delays, but are not directly applicable to distributed delay problems. We show how to reformulate the threshold delay problem as a discrete delay DDE, with the delay as an extra variable, by differentiating the threshold condition. This allows initial value problems to be solved in Matlab using ddesd. We also describe two implementations of the threshold delay DDE in DDE-BIFTOOL to perform numerical bifurcation analysis. One approach enables us to determine steady state bifurcations via a correction of characteristic values from a modified but related problem with discrete delays. The second approach involves introducing dummy constant delays which are used to discretize the integral in the threshold condition. This allows the threshold delay to be solved for directly within DDE-BIFTOOL, and enables the full functionality of DDE-BIFTOOL, so we can compute periodic orbits as well as steady states, along with their stability and bifurcations. We illustrate these techniques on a gene regulation model where the delay, driven by a transport process, is implicitly defined by a threshold condition. The speed of the transport process depends on the state of the system making the delay both state-dependent and distributed.

Spatiotemporal Dynamics of a Delayed Viral Infection Model with Two Modes of Transmission and Immune Impairment

In this paper, the dynamical behaviors for a diffusive virus infection model with two general nonlinear incidence functions, CTL immune impairment and discrete delays, are investigated. Cell‐to‐cell transmission plays a more important role in virus spreading in addition to virus‐to‐cell infection. Nonnegativity and boundcedness properties of the solutions are proven to show the well‐posedness of the model. Five reproduction numbers are calculated and verified to be thresholds determining the global properties of the virus model. A significant impact of cell‐to‐cell transmission is that they increase the basic reproduction number. If one neglects either the cell‐to‐cell transmission or virus‐to‐cell infection, the basic reproduction number of the model is underevaluated. Numerical simulations are carried out to illustrate the dynamical behavior of the model.

Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays

<p style='text-indent:20px;'>We consider a class of neutral type Clifford-valued cellular neural networks with discrete delays and infinitely distributed delays. Unlike most previous studies on Clifford-valued neural networks, we assume that the self feedback connection weights of the networks are Clifford numbers rather than real numbers. In order to study the existence of <inline-formula><tex-math id="M1">\begin{document}$ (\mu, \nu) $\end{document}</tex-math></inline-formula>-pseudo compact almost automorphic solutions of the networks, we prove a composition theorem of <inline-formula><tex-math id="M2">\begin{document}$ (\mu, \nu) $\end{document}</tex-math></inline-formula>-pseudo compact almost automorphic functions with varying deviating arguments. Based on this composition theorem and the fixed point theorem, we establish the existence and the uniqueness of <inline-formula><tex-math id="M3">\begin{document}$ (\mu, \nu) $\end{document}</tex-math></inline-formula>-pseudo compact almost automorphic solutions of the networks. Then, we investigate the global exponential stability of the solution by employing differential inequality techniques. Finally, we give an example to illustrate our theoretical finding. Our results obtained in this paper are completely new, even when the considered networks are degenerated into real-valued, complex-valued or quaternion-valued networks.</p>

Implementation of Efficient XOR Arbiter PUF on FPGA With Enhanced Uniqueness and Security

Physically unclonable functions (PUFs) are recently utilized as a promising security solution for authentication and identification of internet of things (IoT) devices. In this article, we present an efficient implementation of XOR Arbiter PUF (XOR APUF) on field-programmable gate arrays (FPGAs). In our work, we incorporated concept of discrete programmable delays logic (PDL) configurations, the obfuscated challenges and temporal majority voting (TMV) before the XOR operation concurrently to enhance uniqueness and security. The derived design has been verified on 25 Xilinx Artix-7 FPGAs and the results are promising with uniqueness of 48.69%, uniformity 50.73% and reliability 99.41%, which significantly improves over previous work into XOR APUF designs. In addition, we also investigate modeling attack resistance of the proposed design against various modelling attacks and the security analysis show that the proposed PUF has lower prediction rate (<65%) when compared to the previous XOR APUF designs.

Synchronization of Multi-Agent Systems under Time-Varying Network via Time-Delay Approach to Averaging

This paper studies the leader-tracking problem of Multi-Agents Systems under a periodic time-varying communication topology, without requiring the connectivity of the network for all t ≥ 0. The case of both state and communication delays is considered. A fully distributed control protocol, along with the constructive time-delay approach to periodic averaging, are combined in order to solve the problem, thus ensuring that a time-dependent switching control rule preserves the input-to-state stability (ISS) of the entire network, despite the presence of disconnected topologies, state and communication delays. The original closed-loop error systems is transformed into a neutral-type system with discrete and distributed delays. ISS analysis of the neutral system employs appropriate Lyapunov-Krasovskii functionals leading to simple ISS conditions in terms of Linear Matrix Inequalities (LMIs), whose solution allows finding upper bounds on small parameter, state and communication delays that preserve ISS. Numerical simulations illustrate the effectiveness of the theoretical results.

Анализ передачи дискретных сообщений в системах мультиплексирования с ортогональным частотным разделением каналов

High data transfer rates are required in modern digital radio transmission systems. But an increase in the data transfer rate leads to a deterioration in the transmission quality and a high BER (bit error rate). OFDM technology, which was introduced in the 1960s, implements all these functions. In this paper, the influence of fading channels on the transmission of information in OFDM systems is investigated. In wireless systems for transmitting discrete messages, the channel parameters greatly affect the transmission quality, as interference, noise and fading affect the useful signal. Fading is a very common phenomenon in wireless communication systems, affecting the phase shift of the signal. In an urban environment, multipath propagation of signals also takes place, since signals propagate along two or more paths before reaching the receiver. Depending on the signal propagation medium and communication conditions, small-scale fading can be distinguished. At the same time, there are significant changes in the amplitude and phase of the signal. Small-scale fading occurs on a smaller scale, comparable to the wavelength of the transmitted signal, and often changes throughout the signal path, and may change one or more times during the transmission time of the symbol, which leads to so-called flat (smooth) and frequency-selective fading. The results obtained show the effect of fading on the digital transmission system (OFDM). The three discrete message delay profiles used represent a low-, medium-, and high-latency propagation environment, respectively. A model of a wireless digital communication system (OFDM) has been built using the SystemVue software.

Robust stabilization for uncertain saturated systems with multiple time delays

&lt;abstract&gt;&lt;p&gt;This paper is concerned with the robust stabilization problem for uncertain saturated linear systems with multiple discrete delays. First of all, a new distributed-delay-dependent polytopic approach is proposed, and a new type of Lyapunov-Krasovskii functional is constructed. Then, by further incorporating some integral inequalities, both stabilization and robust stabilization conditions are proposed in terms of linear matrix inequalities under which the closed-loop systems are asymptotically stable for admissible initial conditions. Finally, a simulation example is given to illustrate the feasibility and advantages of the obtained results.&lt;/p&gt;&lt;/abstract&gt;

Besicovitch almost periodic solutions for fractional‐order quaternion‐valued neural networks with discrete and distributed delays

In this paper, we study a class of fractional‐order quaternion‐valued neural networks with discrete and distributed delays by direct method. We first investigate some properties of Besicovitch almost periodic functions, including the composition theorem with deviating arguments. Then, we obtain the existence and uniqueness of Besicovitch almost periodic solutions for such class of networks by considering an appropriate Banach space and using the contraction mapping principle. Furthermore, by making use of a generalized Gronwall inequality, we gain the finite‐time stability of the Besicovitch almost periodic solution. Even if the network under consideration degenerates into a real‐valued or complex‐valued networks, the results of this paper are still brand new. Finally, we use a numerical example to show the feasibility of our theoretical results.

Dynamics of fuzzy genetic regulatory networks with leakage and mixed delays in doubly-measure pseudo-almost periodic environment

The study of dynamics behaviors of genetic regulatory networks (GRNs) is essential for the understanding of living organisms at both molecular and cellular levels. This paper deals with a class of fuzzy genetic regulatory networks (FGRNs) with time varying-delays in leakage terms, time-varying discrete delays, unbounded distributed delays, and doubly-measure pseudo-almost periodic parameters. Based on the doubly-measure pseudo-almost periodic theory, exponential dichotomy, differential inequality, and the Banach fixed point theorem, we establish some sufficient conditions to support the existence and global exponential stability of doubly-measure pseudo-almost periodic solutions for the considered model. A numerical example along with a graphical illustration are presented to support our main results. The results of this paper are new and extend existing GRNs models using almost periodic and weighted pseudo-almost periodic functions to support a wider range of regulatory processes.

Improved frequency sweeping technique and stability analysis of the second-order consensus protocol with distributed delays

The consensus problem of a general class of second-order multi-agent system with distributed delays is investigated from a new perspective. We propose a consensus protocol with distributed delays to enhance the delay robustness of the system. First, we establish a spectral equivalence of such a system and the one with multiple discrete delays from the stability viewpoint. We then employ a factorization procedure to reduce the computational complexity. Next, we uniquely use the Dixon resultant theory to determine the exact upper bound of the imaginary spectra, leading to the kernel and offspring hypersurfaces (KOH) in the delay domain. Furthermore, the associated standing roots of the system are analyzed to determine the precise stationary root boundaries, which constitute the potential stability changing curves together with the KOH. Finally, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to examine the exact and complete stability of the system.

A DELAY NONAUTONOMOUS PREDATOR–PREY MODEL FOR THE EFFECTS OF FEAR, REFUGE AND HUNTING COOPERATION

Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.