The Magnus Expansion for Periodic Delay Differential Equations

2013 ◽  
Author(s):  
Árpád Takács ◽  
Eric A. Butcher ◽  
Tamás Insperger
Keyword(s):  
Differential Equations ◽  
Continuous Time ◽  
Delay Differential Equations ◽  
Approximation Technique ◽  
Magnus Expansion ◽  
Time Approximation ◽  
Delayed Differential Equations ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential

In this paper, the application of the Magnus expansion on periodic time-delayed differential equations is proposed, where an approximation technique of Chebyshev Spectral Continuous Time Approximation (CSCTA) is first used to convert a system of delayed differential equations (DDEs) into a system of ordinary differential equations (ODEs), whose solution are then obtained via the Magnus expansion. The stability and time response of this approach are investigated on two examples and compared with known results in the literature.

The Chebyshev Spectral Continuous Time Approximation for Periodic Delay Differential Equations

2009 ◽  
Author(s):  
Eric A. Butcher ◽  
Oleg A. Bobrenkov
Keyword(s):  
Differential Equations ◽  
Continuous Time ◽  
Constant Coefficient ◽  
Delay Differential Equations ◽  
Approximation Technique ◽  
Time Approximation ◽  
Mathieu’S Equation ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential

In this paper, the approximation technique proposed in [1] for converting a system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is applied to both constant and periodic systems of DDEs. Specifically, the use of Chebyshev spectral collocation is proposed in order to obtain the “spectral accuracy” convergence behavior shown in [1]. The proposed technique is used to study the stability of first and second order constant coefficient DDEs with one or two fixed delays with or without cubic nonlinearity and parametric sinusoidal excitation, as well as of the delayed Mathieu’s equation. In all the examples, the results of the approximation by the proposed method show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. In particular, the greater accuracy and convergence properties of this method compared to the finite difference-based continuous time approximation proposed recently in [2] is shown.

On the Analysis of Periodic Delay Differential Equations With Discontinuous Distributed Delays

2011 ◽  
Author(s):  
Oleg A. Bobrenkov ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Differential Equations ◽  
Continuous Time ◽  
Delay Differential Equations ◽  
Monodromy Matrix ◽  
Distributed Delay ◽  
Mathieu Equation ◽  
Adaptive Meshing ◽  
Time Approximation ◽  
Periodic Delay ◽  
Delay Differential

In this paper, the analysis of delay differential equations with periodic coefficients and discontinuous distributed delay is carried out through discretization by Chebyshev spectral continuous time approximation (ChSCTA). These features are introduced in the delayed Mathieu equation with discontinuous distributed delay used as an illustrative example. The efficiency of the process of stability analysis is improved by using shifted Chebyshev polynomials for computing the monodromy matrix, as well as the adaptive meshing of the parameter plane. An idea for a method for numerical integration of periodic DDEs with discontinuous distributed delay based on existing MATLAB functions is proposed.

scholarly journals A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Jin ◽  
Houjun Qi ◽  
Zhanjie Li ◽  
Jianxin Han ◽  
Hua Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Spindle Speed ◽  
Multiple Time ◽  
Variable Pitch ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

A State-Space Temporal Finite Element Approach for Stability Investigations of Delay Equations

2009 ◽  
Author(s):  
Firas A. Khasawneh ◽  
Brian P. Mann ◽  
Bhavin Patel
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
State Space ◽  
Delay Differential Equations ◽  
State Space Model ◽  
Element Analysis ◽  
Generalized Framework ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential

This paper describes a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems — systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

Stability of Delay Equations Written as State Space Models

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1067-1085 ◽  
Author(s):  
B.P. Mann ◽  
B.R. Patel
Keyword(s):  
Differential Equations ◽  
State Space ◽  
Delay Differential Equations ◽  
State Space Model ◽  
Element Analysis ◽  
New Approach ◽  
Generalized Framework ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

scholarly journals Stability Threshold for Scalar Linear Periodic Delay Differential Equations

2016 ◽  
Vol 59 (4) ◽  
pp. 849-857
Author(s):  
Kyeongah Nah ◽  
Gergely Röst
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Trivial Solution ◽  
Delay Differential Equations ◽  
Stability Threshold ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Linear Scalar

AbstractWe prove that for the linear scalar delay differential equationwith non-negative periodic coefficients of period P > 0, the stability threshold for the trivial solution is , assuming that b(t+1)−a(t) does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that, in general, r = 0 is not a stability threshold.

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.

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