Critical Value Of Delay Research Articles

A Delay Mathematical Model for the Control of Unemployment

In this paper, we have proposed and analyzed a nonlinear mathematical model for control of unemployment in the developing countries by incorporating time delay in creating new vacancies. In the modeling process, three dynamic variables have been considered, namely, (i) number of unemployed persons, (ii) number of employed persons, and (iii) number of newly created vacancies. The model is studied using stability theory of differential equations. It is found that the model has only one equilibrium, which is stable in absence of delay. It is further shown that this stable equilibrium becomes unstable as delay crosses some critical value. This critical value of delay has been obtained analytically. Further, direction of Hopf bifurcation and stability of the bifurcating periodic solutions are studied by applying the normal form theory and the center manifold theorem. Numerical simulation of the model has been carried out to illustrate the analytical results.

Critical simple imaginary characteristic roots and invariance properties for quasipolynomials: A missing link

This paper addresses some properties of simple characteristic roots of quasipolynomials including commensurate delays. Although such a problem seems easy and was largely treated in the literature, to the best of the authors' knowledge, the invariance and the ultimate stability properties have not been fully investigated. In this paper, we propose a new frequency-sweeping framework, which simultaneously considers the Taylor series of the critical imaginary roots as well as the Puiseux series of singular points of the frequency-sweeping curves. Through analyzing the algebraic properties of the corresponding frequency-sweeping curves, we are able to completely characterize the invariance property of the simple imaginary roots with respect to the corresponding critical delay values. As a consequence of the invariance property, the ultimate stability property can be easily derived. Finally, as a byproduct of the approach proposed in the paper, the complete stability for time-delay systems with only simple imaginary roots can be systematically derived. Some illustrative examples complete the presentation.

Critical Delays in Clock Distribution Circuits

As the distribution of clock signals between the nodes of a network became a critical operational requisite, the transmission delays have to be studied because they affect the accurate recovering of the time basis. In this work, the critical delay value is calculated considering simplifications compatible with practical situations. The main contribution is to consider the dissipative terms in the node equations, expressing critical delays, depending on the time constant of the dissipation.

Constructive method of linear systems with delay stability analysis

In this paper, the finite constructive method for linear differential-difference systems stability and instability analysis is proposed, and its convergence is proved. This method is based on obtaining of the quadratic lower bound for the quadratic Lyapunov – Krasovskii functionals on some special set of functions. The method proposed is applied to the stability domain construction in a parameter space, and to the critical delay values numerical determination.

Hopf bifurcation analysis of a delayed viral infection model in computer networks

In this paper, a novel computer virus propagation model with dual delays and multi-state antivirus measures is considered. Using theories of stability and bifurcation, it is proven that there exists a critical value of delay for the stability of virus prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the explicit formulas determining the stability and direction of bifurcating periodic solutions are obtained by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to verify the theoretical analysis. The conclusions of this paper can contribute to a better theoretical basis for understanding the long-term actions of virus propagation in networks.

Global Periodic Solutions in a Delayed Predator-Prey System with Holling II Functional Response

We consider a delayed predator-prey system with Holling II functional re- sponse. Firstly, the paper considers the stability and local Hopf bifurcation for a delayed prey-predator model using the basic theorem on zeros of general transcendental function, which was established by Cook etc.. Secondly, special attention is paid to the global ex- istence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu , we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simu- lations supporting the theoretical analysis are given.

A two-parameter geometrical criteria for delay differential equations

In some cases of delay differential equations (DDEs), a delay-dependant coefficient is incorporated into models which takes the form of a function of delay quantity. This brings forth frequent stability-switch phenomena. A geometrical stability criterion is developed on the two-parameter plane for analyzing Hopf bifurcations of equilibria. It is shown that the increasing direction of parameter $\sigma$ would confirm bifurcation directions (from stable one to unstable one, or whereas) at the critical delay values. These lead to the definite partition of stable and unstable regions on the $(\sigma-\tau)$ plane. Several examples are given to illustrate how to use this method to detect both Hopf and double Hopf bifurcations.

Dynamics in three cells with multiple time delays

We consider systems of delay differential equations representing the models containing three cells with any time-delayed connections. Global stability, delay-independent and delay-dependent local stability are studied, the existence of local and global periodic solutions is investigated. We give the stability conditions, respectively, and show that the local periodic solutions can be extended globally after certain critical values of delay.

Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in lin ear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Delay induced Hopf bifurcation in a simplified network congestion control model

Abstract This paper studies delay induced Hopf bifurcation in a simplified network congestion control model. It is shown that there exists a critical value of delay for the stability of the network. When the value of feedback delay passes through the critical value, the system will lose its stability and Hopf bifurcation will occur. The stability and direction of the bifurcating periodic solutions are determined by perturbation procedure. A numerical example is also worked out to verify the theoretical analysis.

Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system

We consider a delayed predator–prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.

On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion

In this paper, we first study the distribution of the zeros of a third degree exponential polynomial. Then we apply the obtained results to a delay model for the control of testosterone secretion. It is shown that under certain assumptions on the coefficients the steady state of the delay model is asymptotically stable for all delay values. Under another set of conditions, there is a critical delay value, the steady state is stable when the delay is less than the critical value and unstable when the delay is greater than the critical value. Thus, oscillations via Hopf bifurcation occur at the steady state when the delay passes through the critical value. Numerical simulations are presented to illustrate the results.

Additional dynamics in transformed time-delay systems

In studying the stability of time delay systems, many published results use a transformation to transform a system with single time delay to a system with distributed delay. In this article, the inherent limitations of such approaches are studied. Specifically, it is shown that such a transformation incurs additional dynamics that can be characterized by appropriate additional eigenvalues. The critical delay values when such additional eigenvalues cross the imaginary axis can be explicitly calculated. If the smallest of such delays is less than the stability delay limit of the original system, then any stability criteria obtained using such transformation will be conservative. Some examples are also included.

Solution moment stability in stochastic differential delay equations.

We study the behavior of the first and second solution moments for linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. In the presence of additive noise (white or colored), the stability domain of both moments is identical to that of the unperturbed system. When these moments lose stability, there is a Hopf bifurcation and the first moment oscillates with a period identical to the solution of the unperturbed equation, while the oscillation period of the second moment is exactly one half the period of the unperturbed solution and the first moment. When perturbations are of the parametric (or multiplicative) type and white noise is assumed, under the It\^o interpretation the first moment of the solution preserves properties of the solution of the deterministic equation, while the behavior of the second moment depends on the amplitude of the stochastic perturbation. The critical delay value at which the second moment loses stability and becomes oscillating is derived, and it is less than the critical delay for the first moment. Under the Stratonovich interpretation, quite different properties were observed for the moment equations, namely, various critical values of the delay and period of oscillations. For the case of parametric colored noise perturbations, sufficient (p-stability) conditions are derived which are independent of the value of delay, and it is shown that colored noise has a stabilizing effect with respect to white noise.