Stability Of Delay Differential Equations Research Articles

Data Driven Approach to Determine Linear Stability of Delay Differential Equations Using Orthonormal History Functions

Abstract Delay differential equations (DDEs) appear in many applications, and determining their stability is a challenging task that has received considerable attention. Numerous methods for stability determination of a given DDE exist in the literature. However, in practical scenarios it may be beneficial to be able to determine the stability of a delayed system based solely on its response to given inputs, without the need to consider the underlying governing DDE. In this work, we propose such a data-driven method, assuming only three things about the underlying DDE: (i) it is linear, (ii) its coefficients are either constant or time-periodic with a known fundamental period, and (iii) the largest delay is known. Our approach involves giving the first few functions of an orthonormal polynomial basis as input, and measuring/computing the corresponding responses to generate a state transition matrix M, whose largest eigenvalue determines the stability. We demonstrate the correctness, efficacy and convergence of our method by studying four candidate DDEs with differing features. We show that our approach is robust to noise, thereby establishing its suitability for practical applications, wherein measurement errors are unavoidable.

Solution Stability of Delay Differential Equations in Banach Spaces

Solution Stability of Delay Differential Equations in Banach Spaces

Mild solutions, variation of constants formula, and linearized stability for delay differential equations

The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.

ON EXPONENTIAL STABILITY OF A CLASS OF DELAY DIFFERENTIAL EQUATIONS IN NEOCLASSICAL GROWTH MODELS

In this paper, the problem of global exponential stability of delay differential equations in a neoclassical growth model is investigated by comparison techniques via differential inequalities.

Research on online advertising attention evaluation decision based on the stability of delay differential equations and Hopf bifurcation analysis

<abstract> This paper uses the stability of the delay differential equation to study its impact on online advertising, helps analyze Hopf branch characteristics in a big data environment, helps companies make online advertising decisions, and maximizes the benefits of product sales. The thesis fully considers various factors such as advertising volume, advertising schedule, and advertising investment level, discusses the singularity types of the advertising delay differential equation, and gives the best decision for advertising investment.The stability of the time-lag differential equation studied in this paper is to study its impact on online advertising, to help analyze the Hopf branch characteristics in the big data environment, and to help companies make online advertising decisions. structure of this article is also from the amount of advertising, the time of advertising, Advertising investment level gradually expands with a certain degree of continuity. </abstract>

A pseudo-spectrum based characterization of the robust strong H-infinity norm of time-delay systems with real-valued and structured uncertainties

Abstract This paper provides a mathematical characterization of the robust (strong) H-infinity norm of an uncertain linear time-invariant system with discrete delays in terms of the robust distance to instability of an associated characteristic matrix. The considered class of uncertainties consists of real-valued, structured, Frobenius norm-bounded matrix uncertainties that act on the coefficient matrices. The robust H-infinity norm, defined as the worst-case value of the H-infinity norm over all admissible uncertainty values, is an important measure of robust performance to quantify the worst-case disturbance rejection of an uncertain dynamical system. For the considered system class, this robust H-infinity norm is however a fragile measure, as for a particular instance of the uncertainties, the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the uncertainties on the system matrices and infinitesimal delay perturbations. This quantity is a continuous function of both the elements of the system matrices and the delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the robust distance to instability of an associated characteristic matrix. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems.

Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps

Delay differential equation is considered under stochastic perturbations of the type of white noise and Poisson's jumps. It is shown that if stochastic perturbations fade on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic differential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.

Analytic and numerical stability of delay differential equations with variable impulses

A stability theory of analytic and numerical solutions to linear impulsive delay differential equations(IDDEs) is established. The stability results in existing literature are extended to IDDEs with variable impulses. A convergent numerical process is proposed to calculate numerical solutions to IDDEs with variable impulses. Convergence and stability of the numerical solutions are studied in the paper. Numerical experiments are given in the end to confirm the conclusion.

Stability of delay differential equations in the sense of Ulam on unbounded intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

On the gamma-logistic map and applications to a delayed neoclassical model of economic growth

In this work, we study the stability properties of a delay differential neoclassical model of economic growth, based on the original model proposed by Solow (Q J Econ 70:65–94, 1956). We consider a logistic-type production function, which comes from combining a Cobb–Douglas function and a linear pollution effect caused by increasing concentrations of capital. The difference between the production function and the classical logistic map comes from the presence of a parameter $$\gamma \in (0,1)$$ in the exponent of one factor. We call this new function the gamma-logistic map. Our main purpose is to obtain sharp global stability conditions for the positive equilibrium of the model and to study how the stability properties of such equilibrium depend on the relevant model parameters. This study is developed by using some properties of the gamma-logistic map and some well-known results connecting stability in delay differential equations and discrete dynamical systems. Finally, we also compare the obtained results with the ones written in related articles.

About stability of delay differential equations with square integrable level of stochastic perturbations

The long term behavior of solutions of stochastic delay differential equations with a fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an asymptotically stable deterministic system remains to be an asymptotically stable (in mean square).

Stability of delay differential equations via delayed matrix sine and cosine of polynomial degrees

In this paper, we study the finite time stability of delay differential equations via a delayed matrix cosine and sine of polynomial degrees. Firstly, we give two alternative formulas of the solutions for a delay linear differential equation. Secondly, we obtain a norm estimation of the delayed matrix sine and cosine of polynomial degrees, which are used to establish sufficient conditions to guarantee our finite time stability results. Meanwhile, a numerical example is presented demonstrating the validity of our theoretical results. Finally, we extend our study to the same issue of a delay differential equation with nonlinearity by virtue of the Gronwall inequality approach.

Numerical radius for the asymptotic stability of delay differential equations

In this note, we make use of the numerical radius to investigate the asymptotic stability of linear systems of delay differential equations. We present examples to illustrate our assumption is weaker than the classical ones.

Global exponential stability of delay difference equations with delayed impulses

This paper investigates the global exponential stability of delay difference equations with delayed impulses. By virtue of Lyapunov functions together with Razumikhin technique, a number of global exponential stability criteria are provided. Both the stability results that impulses act as perturbation and the stability results that impulses act as stabilizer are obtained. Some examples are also presented to illustrate the effectiveness of the obtained results. It should be noted that it is the first time that the Razumikhin type exponential stability results for delay difference equations with delayed impulses are given.

Construction of high-order Runge–Kutta methods which preserve delay-dependent stability of DDEs

This paper is concerned with the construction of some high-order Runge–Kutta methods, which preserve delay-dependent stability of delay differential equations. The methods of the first kind are developed by extending the ideas of Brugnano et al., while the methods of the second kind are developed according to the structure of the stability matrix. We show that the derived methods are τ(0)-stable for delay differential equations. Meanwhile, the Runge–Kutta methods can own the same order of the accuracy as the Radau methods or Gauss methods if the parameters are adequately defined. These results not only improve the order of accuracy of the methods investigated by Huang, but also open an interesting route of finding new τ(0)-stable Runge–Kutta methods. Finally, numerical experiments are proposed to illustrate the theoretical results.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Delays

In this paper, we develop Galerkin approximations for determining the stability of delay differential equations (DDEs) with time periodic coefficients and time periodic delays. Using a transformation, we convert the DDE into a partial differential equation (PDE) along with a boundary condition (BC). The PDE and BC we obtain have time periodic coefficients. The PDE is discretized into a system of ordinary differential equations (ODEs) using the Galerkin method with Legendre polynomials as the basis functions. The BC is imposed using the tau method. The resulting ODEs are time periodic in nature; thus, we resort to Floquet theory to determine the stability of the ODEs. We show through several numerical examples that the stability charts obtained from the Galerkin method agree closely with those obtained from direct numerical simulations.

Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays

Delay differential equations (DDEs) are infinite-dimensional systems, therefore analyzing their stability is a difficult task. The delays can be discrete or distributed in nature. DDEs with distributed delays are referred to as delay integro-differential equations (DIDEs) in the literature. In this work, we propose a method to convert the DIDEs into a system of ordinary differential equations (ODEs). The stability of the DIDEs can then be easily studied from the obtained system of ODEs. By using a space-time transformation, we convert the DIDEs into a partial differential equation (PDE) with a time-dependent boundary condition. Then, by using the Galerkin method, we obtain a finite-dimensional approximation to the PDE. The boundary condition is incorporated into the Galerkin approximation using the Tau method. The resulting system of ODEs will have time-periodic coefficients, provided the coefficients of the DIDEs are time periodic. Thus, we use Floquet theory to analyze the stability of the resulting ODE systems. We study several numerical examples of DIDEs with different kernel functions. We show that the results obtained using our method are in close agreement with those existing in the literature. The theory developed in this work can also be used for the integration of DIDEs. The computational complexity of our numerical integration method is O(t), whereas the direct brute-force integration of DIDE has a computational complexity of O(t2).

Hyers–Ulam stability of delay differential equations of first order

In this paper, we establish the Hyers–Ulam stability of delay differential equations in the form of with a Lipschitz condition on a bounded and closed interval by using successive approximation method.

Galerkin–Arnoldi algorithm for stability analysis of time-periodic delay differential equations

We present an algorithm for determining the stability of delay differential equations (DDEs) with time-periodic coefficients and time-periodic delays. The DDEs are first posed as an equivalent system of partial differential equations (PDEs) along with a nonlinear boundary condition. A Galerkin approximation is then employed to discretize the PDEs into a set of time-periodic ordinary differential equations (ODEs). Finally, we apply a modified version of the Arnoldi algorithm to extract the dominant eigenvalues of the Floquet transition matrix without computing the entire matrix, thereby reducing the required number of integrations of the ODE system. Five numerical examples demonstrate that our modified Arnoldi algorithm provides reliable approximations of the dominant eigenvalues of the Floquet transition matrix, and does so with substantially less computational effort than the classical Floquet method. The stability charts and bifurcation diagrams generated using our Galerkin–Arnoldi method clearly demonstrate the utility of this approach for establishing the stability of a system of DDEs.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.