Synchronizing time delay systems using variable delay in coupling

We present a mechanism for synchronizing time delay systems using one way coupling with a variable delay in coupling that is reset at finite intervals. We present the analysis of the error dynamics that helps to isolate regions of stability of the synchronized state in the parameter space of interest for single and multiple delays. We supplement this by numerical simulations in a standard time delay system like Mackey Glass system. This method has the advantage that it can be adjusted to be delay or anticipatory in synchronization with a time which is independent of the system delay. We demonstrate the use of this method in communication using the bi channel scheme. We show that since the synchronizing channel carries information from transmitter only at intervals of reset time, it is not susceptible to an easy reconstruction.     

Parameter identification in periodic delay differential equations with distributed delay

In this study, a parameter identification approach for identifying the parameters of a periodic delayed system with distributed delay is introduced based on time series analysis and spectral element analysis. Using this approach the parameters of the distributed delayed system can be identified from the time series of the response of the system. The experimental or numerical data of the response is examined with Floquet theory and time series analysis techniques to estimate a reduced order dynamics, or truncated state space to identify the Floquet multipliers. Parameter identification is then completed using a dynamic map developed for the assumed model of the system which can relate the Floquet multipliers to the unknown parameters in the model. The parameter identification technique is validated numerically for first and second order delay differential equations with distributed delay.     

Oscillation of first order delay differential equations with distributed delay

In this paper, we obtain some oscillation criteria for the first order delay differential equation with distributed delay

     

Moment boundedness of linear stochastic delay differential equations with distributed delay

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.     

A condition on delay for differential equations with discrete state-dependent delay

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70 (11) (2009) 3978–3986] is developed. We propose and study an analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions C. The dynamical system is constructed in C and the existence of a compact global attractor is proved.     

Stochastic delay Lotka-Volterra model

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model into the Itô form , and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.     

Fuzzy delay differential equations

In this paper, we prove a local existence and uniqueness result for fuzzy delay differential equations driven by Liu process. We also establish continuous dependence of solution with respect to initial data.     

Balancing with reflex delay

We consider the simplest possible model describing the “man-machine” system when somebody places the end of a stick on his fingertip, and tries to move the lowest point of the stick in a way that its upper position should be stable. The stability analysis of the mechanical model provides surprisingly simple and clear results in spite of the fact that modelling the time delay makes the corresponding differential equation infinite dimensional in mathematical sense. The calculated critical delay of reflexes and the frequency of the nonlinear vibrations arising as a result of a Hopf bifurcation at the limit of linear stability shows good agreement with experimental observations. By means of the above results, we also get some in-view into the work of the organ called “labyrinthus” in the inner ear which helps in self-balancing of the human body even when our eyes are closed.     

A delay decomposition approach to stability analysis of neutral systems with time-varying delay

This paper presents novel stability criteria for neutral systems with time-varying delay. By developing a delayed decomposition approach, information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs). Then, based on the Lyapunov method, delay-dependent stability criteria are devised by taking the relationship between terms in the Leibniz-Newton formula into account. Criteria are derived in terms of LMIs, which can be easily solved by using various convex optimization algorithms. Three illustrative numerical examples are given to show less conservatism of our obtained results and the effectiveness of the proposed method.     

Stability of finite-dimensional and infinite-dimensional nonlinear delay systems, independent of the delay

The stability of nonlinear delay systems is considered. Generalconditions on pseudo-linear finite- and infinite-dimensionaldelay systems are given for stability independent of the delay.     

On collocation methods for delay differential and Volterra integral equations with proportional delay

To compute long term integrations for the pantograph differential equation with proportional delay qt, 0 < q ⩽ 1: y′(t) = ay(t) + by(qt) + f(t), y(0) = y ₀, we offer two kinds of numerical methods using special mesh distributions, that is, a rational approximant with ‘quasi-uniform meshes’ (see E. Ishiwata and Y. Muroya [Appl. Math. Comput., 2007, 187: 741-747]) and a Gauss collocation method with ‘quasi-constrained meshes’. If we apply these meshes to rational approximant and Gauss collocation method, respectively, then we obtain useful numerical methods of order p ^* = 2m for computing long term integrations. Numerical investigations for these methods are also presented.      

脉冲时滞差分方程非振动解的存在性

讨论脉冲时滞差分方程 给出了由时滞差分方程非振动解的存在性刻划出相应的脉冲时滞差分方程的同样性质的一般性脉冲条件。     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.     

A semigroup method for delay equations with relatively bounded operators in the delay term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations.     

Singular control of delay systems

A necessary condition, equivalent to Jacobson's condition, for optimality of a singular control is derived for systems represented by difference-differential equations.     

Pseudospectra and delay differential equations

In this paper, we present a new method for computing the pseudospectra of delay differential equations (DDEs) with fixed finite delay. This provides information on the sensitivity of eigenvalues under arbitrary perturbations of a given size, and hence insight into how stability may change under variation of parameters. We also investigate how differently weighted perturbations applied to the individual matrices of the delayed eigenvalue problem affect the pseudospectra. Furthermore, we compute pseudospectra of the infinitesimal generator of the DDE, from which a lower bound on the maximum transient growth can be inferred. To illustrate our method, we consider a DDE modelling a semiconductor laser subject to external feedback.