Response and Stability Analysis of Periodic Delayed Systems With Discontinuous Distributed Delay

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Oleg A. Bobrenkov ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Stability Analysis ◽  
Continuous Time ◽  
Delay Differential Equations ◽  
Monodromy Matrix ◽  
Distributed Delay ◽  
Mathieu Equation ◽  
Adaptive Meshing ◽  
Delayed Systems ◽  
Time Approximation ◽  
Delay Differential

In this paper, the analysis of delay differential equations with periodic coefficients and discontinuous distributed delay is carried out through discretization by the Chebyshev spectral continuous time approximation (ChSCTA). These features are introduced in the delayed Mathieu equation with discontinuous distributed delay which is used as an illustrative example. The efficiency 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.

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.

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.

Stability Analysis of Fractional Delay Differential Equations by Lagrange Polynomial

2012 ◽  
Vol 500 ◽  
pp. 591-595
Author(s):  
Xiang Mei Zhang ◽  
An Ping Xu ◽  
Xian Zhou Guo
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Numerical Stability ◽  
Mathieu Equation ◽  
Lagrange Polynomial ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The paper deals with the numerical stability analysis of fractional delay differential equations with non-smooth coefficients using the Lagrange collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Lagrange polynomial. Then we study the numerical stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is studied and examined by Lagrange collocation method.

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.

scholarly journals Stability analysis of stochastic delay differential equations with Markovian switching driven by L${\acute{e}}$vy noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yanqiang Chang ◽  
Huabin Chen
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
Markovian Switching ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Almost Surely Exponential Stability ◽  
Delay Differential

<p style='text-indent:20px;'>In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise are studied. The existence and uniqueness of such equations is simply shown by using the Picard iterative methodology. By using the generalized integral, the Lyapunov-Krasovskii function and the theory of stochastic analysis, the exponential stability in <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th(<inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>) for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M4">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.</p>

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