Analysis of a System of Linear Delay Differential Equations

2003 ◽  
Vol 125 (2) ◽  
pp. 215-223 ◽  
Author(s):  
Farshid Maghami Asl ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Complete Solution ◽  
Forced Response ◽  
Necessary Condition ◽  
Linear Delay ◽  
The Stability ◽  
The Individual ◽  
Delay Differential ◽  
Lambert Functions

A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different examples are studied, and the results are presented. The new approach is applied to obtain the stability regions for the individual modes of the linearized chatter problem in turning. The results present a necessary condition to the stability in chatter for the whole system, since it only enables the study of the individual modes, and there are an infinite number of them that contribute to the stability of the system.

Solution of Systems of Linear Delay Differential Equations Via Lambert Functions

2005 ◽  
Author(s):  
Farshid Maghami Asl ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Complete Solution ◽  
Forced Response ◽  
System Stability ◽  
Necessary Condition ◽  
Linear Ordinary Differential Equations ◽  
Linear Delay ◽  
Delay Differential ◽  
Lambert Functions

A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity to the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear DDE’s in matrix form. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Results are presented for stability criteria for the individual modes, free response, and forced response in the context of specific examples. This new approach is also applied to the problem of chatter stability in a machining operation on a lathe. The results, since they are only for individual modes, and there are an infinite number of them, represent a necessary condition for system stability.

A New Look at the Stability Analysis of Delay-Differential Equations

2005 ◽  
Author(s):  
Tama´s Kalma´r-Nagy
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Delay Differential Equations ◽  
Linear Delay ◽  
The Laplace Transform ◽  
Method Of Steps ◽  
The Difference ◽  
The Stability ◽  
Delay Differential

It is shown that the method of steps for linear delay-differential equations combined with the Laplace-transform can be used to determine the stability of the equation. The result of the method is an infinite dimensional difference equation whose stability corresponds to that of the transcendental characteristic equation. Truncations of this difference equation are used to construct numerical stability charts. The method is demonstrated on a first and second order delay equation. Correspondence between the transcendental characteristic equation and the difference equation is proved for the first order case.

Stability Aspects Of Explicit Runge-Kutta Method For Delay Differential Equations

2012 ◽  
Author(s):  
Fudziah Ismail ◽  
San Lwin Aung ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Linear Delay ◽  
Different Types ◽  
Regions Of Stability ◽  
Runge Kutta ◽  
The Stability ◽  
Delay Differential

Persamaan pembezaan lengah linear (PPL) diselesaikan dengan kaedah Runge–Kutta menggunakan interpolasi yang berbeza bagi penghampiran sebutan lengahnya. Polinomial kestabilannya diterbitkan dan rantau kestabilannya dipersembahkan. Kata kunci: Runge-Kutta, persamaan pembezaan lengah, kestabilan, interpolasi The linear delay differential equations (DDEs) are solved by Runge–Kutta method using different types of interpolation to approximate the delay terms. The stability polynomials are derived and the respective regions of stability are presented. Key words: Runge-Kutta, delay differential equations, stability, interpolation

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear 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 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.

Asymptotic behaviour, nonoscillation and stability in periodic first-order linear delay differential equations

1998 ◽  
Vol 128 (6) ◽  
pp. 1371-1387 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Characteristic Equation ◽  
Trivial Solution ◽  
Delay Differential Equations ◽  
Order Linear ◽  
First Order ◽  
Linear Delay ◽  
The Stability ◽  
Delay Differential

First-order scalar linear delay differential equations with periodic coefficients and constant delays are considered, where the coefficients have a common period and the delays are multiples of this period. A basic asymptotic criterion is given. Moreover, some results on the nonoscillation and on the stability of the trivial solution are obtained. An equation, which is in a sense the characteristic equation, plays an important role in establishing the results of the paper.

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

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
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.

scholarly journals An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Süleyman Cengizci
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Numerical Procedure ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Convection Term ◽  
Linear Delay ◽  
Delay Differential

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

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