Class Of Delay Differential Equations Research Articles

State Dependent Delay Maps: Numerical Algorithms and Dynamics of Projections

This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a function of the state variable. A delay map with constant delays can be rewritten explicitly as a discrete time dynamical system on an appropriate function space, and a delay map with small state dependent terms can be viewed as a “non-autonomous” perturbation. We develop a fixed point formulation for the Cauchy problem of such perturbations, and under appropriate assumptions obtain the existence of forward iterates of the map. The proof is constructive and leads to numerical procedures which we implement for illustrative examples, including the cubic Ikeda and Mackey-Glass systems with constant and state-dependent delays. After proving a local convergence result for the method, we study more qualitative/global convergence issues using data analytic tools for time series analysis (dimension and topological measures derived from persistent homology). Using these tools we quantify the convergence of the dynamics in the finite dimensional projections to the dynamics of the infinite dimensional system.

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

Basins of attraction and paired Hopf bifurcations for delay differential equations with bistable nonlinearity and delay-dependent coefficient

We consider a class of delay differential equations with bistable nonlinearity, in which the trivial equilibrium may coexist with two positive equilibria. Despite the difficulty caused by delay and bistable nonlinearity, we give a rather complete description on the dynamics including global stability, semi-stability, bistability and Hopf bifurcation. For the case where the stable trivial equilibrium coexists with a stable positive equilibrium, we obtain two delay-dependent intervals as subsets of basins of attraction of two stable equilibria. These subsets are sharp in some sense. Using delay as the bifurcation parameter, we analytically show that the number of local Hopf bifurcation values is finite and these local Hopf bifurcation values are neatly paired. A Nicholson's blowflies equation with Allee effect is used to illustrate our general results. Through this example, we show that delay can induce stability switches, symmetric transitions among multitype bistability and robust phase-transitions for long transients.

Fitted Parameter Exponential Spline Method for Singularly Perturbed Delay Differential Equations with a Large Delay

In this paper, we present a new computational method based on an exponential spline for solving a class of delay differential equations with a negative shift in the differentiated term. When the shift parameter is O ε , the proposed method works well and also controls the oscillations in the solution’s layer region. To accomplish this, we included a parameter in the proposed numerical scheme that is based on a special type of mesh, and the parameter is evaluated using the theory of singular perturbation. Maximum absolute errors and convergences of numerical solutions are tabulated to demonstrate the efficiency of the proposed computational method and to support the convergence analysis of the presented scheme.

Stability and feedback stabilizability of delay periodic differential equations with pairwise permutable matrix functions

Abstract Representation of solutions of delayed differential equations with multiple delays and periodic coefficients is established. Consequently, results on stabilizability of weakly delayed closed-loop systems and stability of non-weakly delayed periodic systems are proved. The stabilizability result is an extension of the classical Brunovský theorem for linear periodic systems of ordinary differential equations to a class of delay differential equations with pairwise permutable matrix functions.

A New Result for ψ‐Hilfer Fractional Pantograph‐Type Langevin Equation and Inclusions

In this paper, we deal with the existence and uniqueness of solution for ψ‐Hilfer Langevin fractional pantograph differential equation and inclusion; these types of pantograph equations are a special class of delay differential equations. The existence and uniqueness results are obtained by making use of the Krasnoselskii fixed‐point theorem and Banach contraction principle, and for the inclusion version, we use the Martelli fixed‐point theorem to get the existence result. In the end, we are giving an example to illustrate our results.

Collocation method based on radial basis functions via symmetric variable shape parameter for solving a particular class of delay differential equations

In this article, we use the collocation method based on the radial basis functions with sym- metric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional- differential equations with proportional delays. We used Gaussian radial basis functions with SVSP. Using non uniform collocation points, we achieved a system and solving this system yielded the prob- lem solutions. Several examples are given to illustrate the efficiency and accuracy of the introduced method in comparison with the same method with the constant shape parameter (CSP) as well as other analytical and numerical methods. Comparison of the obtained numerical results shows the considerable superiority of the collocation method based on RBFs with SVSP in accuracy and convergence over the collocation method based on the RBFs with CSP and other analytical and numerical methods for delay differential equations (DDEs). Finally, numerical rate of convergence analysis of the numerical approximation was also obtained. It is observed that by comparing be- tween the obtained ROC values of error norms by the SVSP and CSP method, SVSP results were considered acceptable.

Optimal control of an avian influenza model with multiple time delays in state and control variables

In this paper, we consider an optimal control model governed by a class of delay differential equation, which describe the spread of avian influenza virus from the poultry to human. We take three control variables into the optimal control model, namely: slaughtering to the susceptible and infected poultry ( \begin{document}$ u_{1}(t) $\end{document} ), educational campaign to the susceptible human population ( \begin{document}$ u_{2}(t) $\end{document} ) and treatment to infected population ( \begin{document}$ u_{3}(t) $\end{document} ). The model involves two time delays that stand for the incubation periods of avian influenza virus in the infective poultry and human populations. We derive first order necessary conditions for existence of the optimal control and perform several numerical simulations. Numerical results show that different control strategies have different effects on controlling the outbreak of avian influenza. At the same time, we discuss the influence of time delays on objective function and conclude that the spread of avian influenza will slow down as the time delays increase.

Periodic solutions to certain classes of third order delay differential equations

In this paper, existence of unique periodic solutions for two classes of third order delay differential equations are considered. The two equations are expressed in their integral equivalents with which suitable Green’s function is constructed and its associated properties stated. The main tool adopted to establish the existence of unique periodic solutions to the classes of delay differential equations is the Krasnoselskii’s fixed point theorem due to its applicability to the combination of a compact and contraction mappings which occur in dealing with perturbed differential operators.

A Malmquist Type Theorem for a Class of Delay Differential Equations

We show that if the following delay differential equation of rational coefficients $$\begin{aligned} w^k(z)\sum _{\mu =1}^se_\mu (z)w(z+c_\mu )+a(z)\frac{w^{(n)}(z)}{w(z)}= \frac{\sum _{i=0}^pa_i(z)w^i}{\sum _{j=0}^qb_j(z)w^j} \end{aligned}$$ admits a transcendental entire solution w of hyper-order less than one, then it reduces into a delay differential equation of rational coefficients $$\begin{aligned} w^k(z)\sum _{\mu =1}^se_\mu (z) w(z+c_\mu )+a(z)\frac{w^{(n)}(z)}{w(z)}=\frac{1}{w(z)}\sum _{i=0}^{k+2}A_{i}(z)w^i(z), \end{aligned}$$ which improves some known theorems obtained most recently by Zhang and Huang. Some examples are constructed to show that our results are accurate.

Over- and Underapproximating Reach Sets for Perturbed Delay Differential Equations

This article explores reach set computations for perturbed delay differential equations (DDEs). The perturbed DDEs of interest in this article is a class of DDEs whose dynamics are subject to perturbations, and their solutions feature the local homeomorphism property with respect to initial states. Membership in this class of perturbed DDEs is determined by conducting sensitivity analysis of solution mappings with respect to initial states to impose a bound constraint on the time-lag term. The homeomorphism property of solutions to such class of perturbed DDEs enables us to construct over- and underapproximations of reach sets by performing reachability analysis on just the boundaries of their permitted initial sets, thereby permitting an extension of reach set computation methods for ordinary differential equations to perturbed DDEs. Three examples demonstrate the performance of our approach.

Эффективная программная реализация численных методов решения жестких систем дифференциальных уравнений с запаздывающим аргументом

Системы уравнений с запаздываниями широко применяются в различных областях современного математического моделирования. В ходе разработки структуры математической модели и идентификации ее параметров приходится многократно решать задачу Коши для подобных систем. В случае высокой размерности системы, а также при условии жесткости задачи численное решение уравнений с запаздываниями может требовать значительных вычислительных и временных затрат. Таким образом, разработка и реализация эффективных алгоритмов численного решения различных классов уравнений с запаздывающими аргументами является актуальной задачей. В настоящей статье представлена модифицированная версия программного комплекса DIFSUBDEL, в которой реализованы методы численного решения дифференциальных уравнений с запаздываниями на основе линейных многошаговых методов. Переработанная версия разработана с применением принципов структурного программирования и является значительно более удобной в эксплуатации, чем исходная, а также обладает свойством потокобезопасности, что позволяет использовать комплекс в качестве блока в системах, основанных на технологиях параллельного программирования с общей памятью. Был проведен сравнительный анализ производительности переработанной системы DIFSUBDEL c другими существующими программными реализациями численных методов решения дифференциальных уравнений с запаздыванием и показана ее эффективность. The systems of delay differential equation are widely used in modern mathematical modeling. The process of mathematical model development and identification requires the repeated solution of initial value problems for such systems. The numerical solution of delay differential equations may be computationally expensive, especially when the problem is stiff and high-dimensional. Therefore, it is important to develop and implement the efficient algorithms for the numerical solution of different classes of delay differential equations. In this paper, a new implementation of DIFSUBDEL program package for the numerical solution of delay differential equations based on linear multistep methods is discussed. The modified version is based on structured programming principles to make the program thread safe and user-friendly. The modified program performance is compared with the existing implementations of numerical methods for the solution of delay differential equations.

Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions

In this article we study a class of delay differential equations with infinite delay in weighted spaces of uniformly continuous functions. We focus on the integrated semigroup formulation of the problem and so doing we provide a spectral theory. As a consequence we obtain a local stability result and a Hopf bifurcation theorem for the semiflow generated by such a problem.

C 1 -smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in L p

The objective of this paper is to clarify the relationship between the $C^1$-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class of DDEs which include a constant discrete delay. The problem of $C^1$-smooth dependence is fundamental from the viewpoint of the theory of differential equations. However, the above mentioned relationship is not obvious because the corresponding functional differential equations have the less regularity with respect to the delay parameter. In this paper, we prove that the $C^1$-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in $L^p$ plays an important role.

Consensus of a two-agent opinion dynamical system with processing delay

A sufficient condition is obtained for a two-agent opinion dynamical system with processing delay to experience unconditional consensus. To this end, the dynamics proposed is transformed into a general class of delay differential equation and asymptotic stability of the origin is then analyzed. It is also shown that increasing delay can prolong the time needed for the system to realize consensus and even induce a Hopf bifurcation.

On the Number of Periodic Solutions of Delay Differential Equations

In this paper, we consider the existence and number of periodic solutions for a class of delay differential equations of the form [Formula: see text], based on the Kaplan–Yorke method. Especially, we consider a kind of delay differential equations with [Formula: see text] as a polynomial having parameters and find the number of periodic solutions with period [Formula: see text] or [Formula: see text].

A frequency-domain method for solving linear time delay systems with constant coefficients

In an active control system, time delay will occur due to processes such as signal acquisition and transmission, calculation, and actuation. Time delay systems are usually described by delay differential equations (DDEs). Since it is hard to obtain an analytical solution to a DDE, numerical solution is of necessity. This paper presents a frequency-domain method that uses a truncated transfer function to solve a class of DDEs. The theoretical transfer function is the sum of infinite items expressed in terms of poles and residues. The basic idea is to select the dominant poles and residues to truncate the transfer function, thus ensuring the validity of the solution while improving the efficiency of calculation. Meanwhile, the guideline of selecting these poles and residues is provided. Numerical simulations of both stable and unstable delayed systems are given to verify the proposed method, and the results are presented and analysed in detail.

Permanence in a class of delay differential equations with mixed monotonicity

Permanence in a class of delay differential equations with mixed monotonicity

Oscillation of a linear delay differential equation with slowly varying coefficient

A new sufficient condition is given for the oscillation of all solutions of a linear scalar delay differential equation with slowly varying and uniformly positive coefficient at infinity. Based on the celebrated Myshkis type necessary condition, we show that this new sufficient condition is sharp within the considered class of delay differential equations.

An alternative use of fuzzy transform with application to a class of delay differential equations

ABSTRACTFuzzy transforms (or F-transforms for short) are an approximation technique recently introduced. The main application is referred to image and data compression. There are really few works devoted to the use of F-transform for solving ordinary differential equations. In the present paper, an F-transform-based Picard-like numerical scheme is proposed in order to solve a class of delay differential equations. For linear cases, the proposed approach leads to a non-recursive approximate solution by means of operational matrices and vectors of known quantities. Numerical results show the good performance of the proposed method against known solutions.