Delay-dependent Coefficients Research Articles

Stability analysis of systems with delay-dependent coefficients and commensurate delays

Stability analysis of systems with delay-dependent coefficients and commensurate delays

Further study on the crossing curves in two-delay differential equations with delay-dependent coefficients

The crossing curves method, which allows delays to vary simultaneously, is generalized to the scenario that four terms exist in the characteristic equations with delay-dependent coefficients. The crossing curves on the two-delay parameter plane are first plotted by our generalized algorithms. The criteria to determine the crossing direction are also given. Finally, an example is provided to support our method and illustrate its fortes.

Dynamics of an HIV infection model with two time delays

In this paper, a mathematical model for HIV-1 infection and immune response is considered, involving two discrete time delays in the intracellular as well as in activation of immune response. Using a recently developed geometric method for studying a class of transcendental equation with two time delays and delay dependent coefficients, we obtain the stability and bifurcation results at the non-trivial equilibrium. In particular, the crossing curves on the two-delays parameter plane can be completely characterized, on which Hopf and double-Hopf bifurcation will take place. In the case of Hopf bifurcation, there exist stability switches, and the direction and stability of delay induced Hopf-bifurcation can be determined using normal form theory and center manifold theorem. These results imply the model will exhibit complex temporal dynamics, such as period oscillations, quasi-periodic solutions, etc. Numerical examples are also carried out to verify these results.

A non-linear approach to Kalecki’s investment cycle

In this paper, looking at some new contributions to the Kalecki business cycle theory, we re-examine his 1935 model concerning the gestation period between orders and deliveries of capital goods. The model gives rise to a delay differential equation (DDE) with delay dependent coefficients depending on the time delay. The model has only one equilibrium point. Proved that a unique stability switch exists, we study the emergence of the Hopf bifurcation and the direction, stability and period of the bifurcating periodic solutions. We derive an explicit formula for determining the properties of the Hopf bifurcation by using the first Lyapunov coefficient.To confirm our analytic results, we consider two types of non-linear functions, all consistent with Kalecki’s hypotheses: two S-shaped functions and one fractional function. Some comments dealing with the economic implications of our analysis are also included.

Algorithms for fast calculation of spectral abscissa for retarded time-delay systems with delay-dependent coefficients

Algorithms for fast calculation of spectral abscissa for retarded time-delay systems with delay-dependent coefficients

Stability and direction of Hopf bifurcations of a cyclical growth model with two-time delays and one-delay dependent coefficients

• We study the impact of two discrete-time delays on the basic Goodwin growth cycle model. • We study a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delay. • Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation. • We analyze the direction and stability of Hopf bifurcation by means of the first Lyapunov coefficient. This paper deals with the impact of two discrete-time delays on the basic Goodwin growth cycle model. The former concerns the existence of a finite time delay for building capital goods as suggested by Kalecki. The latter pertains to the wage lag hypothesis. This is because, taking the current change of the employment rate into account, workers and capitalists bargain new wage periodically. There are no examples in the literature on the Goodwin model of the use of both those lags in order to explore the GDP dynamics. From the analytical point-of-view, what we obtain is a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delays. Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation resulting from the stability analysis and the direction of the Hopf bifurcations. Although the system with no lag displays a stable focus, the introduction of the two lags preserves the stable solution only for particular combinations of parameters and length of the lags. In any other case, instability prevails and regular cycles or chaotic fluctuations emerge. Finally, we provide the analytical results with the necessary economic interpretations.

Delay-induced stability switches in an SIRS epidemic model with saturated incidence rate and temporary immunity

This work considers a time-delayed SIRS epidemic model with temporary immunity and nonlinear incidence rate, where the susceptible host population satisfies the logistic equation and the incidence rate is of saturated form with the susceptible. The time delay represents a period of temporary immunity where disease-recovered individuals return to the susceptible class after a fixed period of time. By analyzing the associated characteristic equation with delay-dependent coefficients and regarding the time lag as the bifurcation parameter, the local stability of the endemic equilibrium is investigated and sufficient conditions for the occurrence of stability switches via Hopf bifurcations are obtained. It is shown that the delay parameter can induce a finite number of stability switches before completely destabilizing the system. Numerical simulations are carried out to illustrate theoretical results.

Insights on Asymptotic Behavior of Characteristic Roots of Quasi-Polynomials with Delay-Dependent Coefficients in Some Ill-Posed Cases

This work focus on the asymptotic behavior of critical roots of linear time-delay systems with delay-dependent coefficients. Our main goal is to derive a simple methodology to characterize some ill-posed cases from the open literature. In order to accomplish such a task, our main tool will be to express the solutions as a convergent power series. More specifically, by means of an appropriate change of variable, the expansion of unbounded solutions is obtained, which allows to derive an appropriate auxiliary solution that serves to analyze the behavior of such solutions. One of the main benefits in adopting such an approach, is that stability analysis in the ill-posed cases can also be covered.

Stability Switches of a Class of Fractional-Delay Systems With Delay-Dependent Coefficients

Stability of a dynamical system may change from stable to unstable or vice versa, with the change of some parameter of the system. This is the phenomenon of stability switches, and it has been investigated intensively in the literature for conventional time-delay systems. This paper studies the stability switches of a class of fractional-delay systems whose coefficients depend on the time delay. Two simple formulas in closed-form have been established for determining the crossing direction of the characteristic roots at a given critical point, which is one of the two key steps in the analysis of stability switches. The formulas are expressed in terms of the Jacobian determinant of two auxiliary real-valued functions that are derived directly from the characteristic function, and thus, can be easily implemented. Two examples are given to illustrate the main results and to show an important difference between the fractional-delay systems with delay-dependent coefficients and the ones with delay-free coefficients from the viewpoint of stability switches.

Stability Analysis of a More General Class of Systems With Delay-Dependent Coefficients

This paper presents a systematic method to analyze the stability of systems with single delay in which the coefficient polynomials of the characteristic equation depend on the delay. Such systems often arise in, for example, life science and engineering systems. A method to analyze such systems was presented by Beretta and Kuang in a 2002 paper, but with some very restrictive assumptions. This paper extends their results to the general case with the exception of some degenerate cases. It is found that a much richer behavior is possible when the restrictive assumptions are removed. The interval of interest for the delay is partitioned into subintervals so that the magnitude condition generates a fixed number of frequencies as functions of the delay within each subinterval. The crossing conditions are expressed in a general form, and a simplified derivation for the first-order derivative criterion is obtained. Illustrative examples are also presented in this paper.

Stability and bifurcation analysis of a delayed innovation diffusion model

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage (time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period (time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Stability Analysis of Systems With Delay-Dependent Coefficients: An Overview

This paper gives an overview of the stability analysis of systems with delay-dependent coefficients. Such systems are frequently encountered in various scientific and engineering applications. Most such analyses are generalization of those on systems with delay-independent coefficients . Therefore an introduction on systems with delay-independent coefficients is also given, with an emphasis on the $\tau $ -decomposition approach. Methods for two key ingredients of this approach are discussed, namely the identification of imaginary characteristic roots with the corresponding delays, and local behavior analysis of these roots as the delay increases through these critical values. For systems with delay-dependent coefficients , we review the methods of analysis for systems with a single delay and commensurate delays, their application to output feedback control and a geometric perspective that establishes a link between systems with and without delay-dependent coefficients. We provide the main ideas of various stability analysis methods and their advantages and limitations. We also present our perspectives on future directions of research on this interesting topic.

Stability Analysis of Control Systems subject to Delay-Difference Feedback

In engineering practice, delay-difference is often used to approximate the derivatives of output signals for feedback control, leading to a closed-loop system with delay both in the states and in the system’s coefficients. In this context, our objective is to find all the delay values contained in some interval that guarantee the exponential stability of the closed-loop system subject to the delay-difference approximation. A method for stability analysis of systems with delay-dependent coefficients developed in our previous work is further extended and applied to tackle the particular form of systems considered in this paper. The proposed stability analysis procedure is illustrated through the design of a mobile-robot path-following controller.

Modeling transcriptional co-regulation of mammalian circadian clock.

The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

Complexity in a predator-prey-parasite model with nonlinear incidence rate and incubation delay

This paper is concerned with a predator-prey-parasite model with nonlinear infection rate and incubation delay. To explore the system dynamics, we study the distribution of roots of the characteristic equation of the Jacobian matrix of the system which has delay-dependent coefficients. The dynamics displayed by the system can exhibit some of the key features observed in the natural systems, such as appearance and disappearance of cycles in succession. It is shown that the switching phenomenon between stable coexistence and oscillatory coexistence of interior equilibrium as well as of predator-free equilibrium is an interplay of three system parameters, viz. the crowding coefficient, the prey consumption rate and the length of incubation delay. We also discuss the stability of the delayed-system when the non-delayed system is assumed to be unstable.

Guaranteed interval of delay for a class of non-linear control systems

A new method is proposed in this article in order to determine a guaranteed interval of time delay in input signal for a class of non-linear control systems with state feedback. This method is based on the stability sufficient condition of a non-linear control system using the Lyapunov direct approach. For this purpose, the dynamic equations of the non-linear input-delayed system are reformulated as a non-linear system without delay, but with delay-dependent coefficients in the state-space description. The control strategy is considered a linear state feedback. Then, by introducing an appropriately related Lyapunov candidate, the stability sufficient condition of the closed-loop system is satisfied, leading to a guaranteed interval of delay according to the state feedback gain. The main results are gathered and proved as a lemma. The effectiveness of this proposed method is investigated by simulating the state feedback control of a non-linear synchronous generator with a delay in input.

Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays

In this paper, a tumor and immune system interaction model consisted of two differential equations with three time delays is considered in which the delays describe the proliferation of tumor cells, the process of effector cells growth stimulated by tumor cells, and the differentiation of immune effector cells, respectively. Conditions for the asymptotic stability of equilibria and existence of Hopf bifurcations are obtained by analyzing the roots of a second degree exponential polynomial characteristic equation with delay dependent coefficients. It is shown that the positive equilibrium is asymptotically stable if all three delays are less than their corresponding critical values and Hopf bifurcations occur if any one of these delays passes through its critical value. Numerical simulations are carried out to illustrate the rich dynamical behavior of the model with different delay values including the existence of regular and irregular long periodic oscillations.

Stability and Bifurcation Analysis in Leukopoiesis Models with Two Delays

We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of Hopf bifurcations which leads to periodic solutions.

Stability Switches and Hopf Bifurcation in a Kaleckian Model of Business Cycle

This paper considers a Kaleckian type model of business cycle based on a nonlinear delay differential equation, whose associated characteristic equation is a transcendental equation with delay dependent coefficients. Using the conventional analysis introduced by Beretta and Kuang (2002), we show that the unique equilibrium can be destabilized through a Hopf bifurcation and stability switches may occur. Then some properties of Hopf bifurcation such as direction, stability, and period are determined by the normal form theory and the center manifold theorem.

Dynamic Properties of the Solow Model with Increasing or Decreasing Population and Time-to-Build Technology

We introduce a time-to-build technology in a Solow model with nonconstant population. Our analysis shows that the population dynamics may be a source of stability switches and Hopf bifurcations. The analytical results are obtained using the recent technique introduced by Beretta and Kuang (2002) in the studying of delayed differential equations with delay-dependent coefficients in characteristic equation. Numerical simulations are performed in order to illustrate the main dynamic features of the model.