Semi‐discretization method for delayed systems

The paper presents an efficient numerical method for the stability analysis of linear delayed systems. The method is based on a special kind of discretization technique with respect to the past effect only. The resulting approximate system is delayed and also time periodic, but still, it can be transformed analytically into a high‐dimensional linear discrete system. The method is applied to determine the stability charts of the Mathieu equation with continuous time delay. Copyright © 2002 John Wiley & Sons, Ltd.     

Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials

This paper presents a new technique for studying the stability properties of dynamic systems modeled by delay‐differential equations (DDEs) with time‐periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the ‘infinite‐dimensional Floquet transition matrix U’. Two different formulas for the computation of the approximate U, whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second‐order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time‐periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance‐modulated turning. The results indicate that this method is an effective way to study the stability of time‐periodic DDEs. Copyright © 2004 John Wiley & Sons, Ltd.     

Two results on stable rapidly oscillating periodic solutions of delay differential equations

In 1999 Ivanov and Losson [A.F. Ivanov and J. Losson, Stable rapidly oscillating solutions in delay differential equations with negative feedback, Differ. Int. Eqns 12 (1999), pp. 811–832] presented a computer assisted proof that a particular delay differential equation (with negative feedback) admits a stable rapidly oscillating periodic solution (ROPS). In this article the delay equation of Ivanov and Losson is embedded in a five-parametric class of differential equations. Conditions on the parameters are given such that the delay equation admits a stable ROPS. Moreover, it is shown that for odd n?>?1 the delay equation admits a stable ROPS with n humps per unit time if the parameters satisfy some explicitly given conditions. The delay equation of Ivanov and Losson satisfies all conditions on the five parameters. This gives an analytic proof and a considerable generalization of the result of Ivanov and Losson. The conditions on the parameters are believed to be sharp in a certain sense. The second result proves part of a conjecture in Stoffer [D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Differ. Eqns 20(1) (2008), pp. 201–238]. For a class of stiff delay differential equations with piecewise constant nonlinearity (positive or negative feedback) and for every n the following holds: if the stiffness parameter is sufficiently large then there are 2a(n) essentially different stable ROPSs with n humps per time unit. a(n) is the number of essentially different binary n-stage shift register sequences.     

Stochastic semi-discretization for linear stochastic delay differential equations

An efficient numerical method is presented to analyze the moment stability and stationary behavior of linear stochastic delay differential equations. The method is based on a special kind of discretization technique with respect to the past effects. The resulting approximate system is a high dimensional linear discrete stochastic mapping. The convergence properties of the method is demonstrated with the help of the stochastic Hayes equation and the stochastic delayed oscillator.     

Robust stability for perturbed discrete systems with time delay

Abstract

In this paper, we present a roubust stability condition for linear perturbed discrete time‐delay systems. The perturbations are treated with highly structured information. A state feedback controller for the aforementioned systems is also introduced. Examples are given to demonstrate the result.     

Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Bifurcations of the periodic stationary solutions of nonlinear time‐periodic time‐delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay‐differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map that describes the evolution over the time period. Fold, flip, and Neimark‐Sacker bifurcations related to the fixed point of this map are analyzed via center manifold reduction and normal form theorems. The analysis unfolds the approximate stability properties and bifurcations of the stationary solution of the delay‐differential equation and, at the same time, allows the approximate computation of the arising period‐1, period‐2, and quasi‐periodic solution branches. The method is demonstrated for the delayed Mathieu‐Duffing equation, and the results are verified by numerical continuation.     

A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time‐periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous‐time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts. Copyright © 2016 John Wiley & Sons, Ltd.     

D‐Stability analysis for discrete delay systems

Abstract

This paper discusses the D‐pole placement problem of discrete time‐delay systems where the delay duration can be any positive integer. One sufficient condition is proposed to insure all the closed‐loop eigenvalues of discrete delay systems be located inside a specific disk D(α, r). Several criteria are used to retain all the eigenvalues of discrete delay systems with structured or unstructured parametric perturbations inside the same disk. Finally, some illustrative examples are given to demonstrate our results.     

Analytical solutions of delay‐differential equations via a time‐partition method

Abstract

A time‐partition method is used to obtain the analytical solutions of delay‐differential equations by the Laplace transformation technique with a special matrix inversion algorithm.     

Asymptotic properties of the spectrum of neutral delay differential equations

Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.     

Almost periodic solutions for impulsive fractional differential equations

In this paper, we consider the existence of almost periodic solutions for impulsive fractional evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, probability density functions, fixed point theorems and the techniques based on fractional calculus. An example is also discussed to illustrate the theory. Some known results are improved and generalized.     

Combining differential quadrature method with simulation technique to solve non‐linear differential equations

Nowadays, most of the ordinary differential equations (ODEs) can be solved by modelica‐based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non‐linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non‐linear problems and how the CM gives more accurate results compared with those of other methods. Copyright © 2008 John Wiley & Sons, Ltd.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability and robustness of time‐delay systems

Abstract

Using some properties of matrix measures and matrix's spectral radius, a new stability criterion for a linear time‐delay system is derived. This result is also extended to interval time‐delay systems.     

互相关法时延估计中的多径抑制方法

浅海信道是一个多径信道,对于基于相关法的延迟估计,多径效应常常会导致相关函数中出现伪峰,伪峰的幅度有时会超过主峰,进而增加了相关峰选取的难度。利用 AR 法滤波对信号中的多径分量加以抑制,从而削弱了伪峰的幅度,提高了延迟估计的准确性。先从理论上论证了线性预测法可以对多径干扰进行抑制,再用仿真信号和实际信号验证了方法的效果。对于仿真信号,在无噪声干扰的条件下,选择合适阶数的滤波器可以使信号中的多径分量被完全抑制;对于正信噪比的实际信号,其中的多径分量在一定程度上被抑制,从而由多径分量产生的相关峰会被明显削弱,减少了峰值位置的跳变。     

Notes on multiple periodic solutions for second-order discrete Hamiltonian system

By a new orthogonal direct sum decomposition E_M = Y⊕Z, where Z is related to Δu_i(i = 1, 2, 3, …, M), and a new functional I(u), the method by Guo and Yu is improved to obtain new multiple periodic solutions with a negativity hypothesis on F for a second-order discrete Hamiltonian system. Moreover, we exhibit an instructive example to make our result more clear, which has not been solved by known results.     

Fourth‐order energy‐preserving locally implicit time discretization for linear wave equations

A family of fourth‐order coupled implicit–explicit time schemes is presented as a special case of fourth‐order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into several regions where different fourth‐order time discretizations are used, chosen among a family of implicit or explicit fourth‐order schemes. The coupling is based on a Lagrangian formulation on the boundaries between the several non‐conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth‐order consistency in time. Numerical results in 1D and 2D for the acoustic and elastodynamics equations illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be extremely penalizing. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step and makes this family of schemes attractive compared with second‐order accurate methods. Copyright © 2015 John Wiley & Sons, Ltd.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section

To improve the ability of the scaled boundary finite element method (SBFEM) in the dynamic analysis of dam–reservoir interaction problems in the time domain, a diagonalization procedure was proposed, in which the SBFEM was used to model the reservoir with uniform cross‐section. First, SBFEM formulations in the full matrix form in the frequency and time domains were outlined to describe the semi‐infinite reservoir. No sediments and the reservoir bottom absorption were considered. Second, a generalized eigenproblem consisting of coefficient matrices of the SBFEM was constructed and analyzed to obtain corresponding eigenvalues and eigenvectors. Finally, using these eigenvalues and eigenvectors to normalize the SBFEM formulations yielded diagonal SBFEM formulations. A diagonal dynamic stiffness matrix and a diagonal dynamic mass matrix were derived. An efficient method was presented to evaluate them. In this method, no Riccati equation and Lyapunov equations needed solving and no Schur decomposition was required, which resulted in great computational costs saving. The correctness and efficiency of the diagonalization procedure were verified by numerical examples in the frequency and time domains, but the diagonalization procedure is only applicable for the SBFEM formulation whose scaling center is located at infinity. Copyright © 2009 John Wiley & Sons, Ltd.