Estimation of Time-Delays Using the Characteristic Roots of Delay Differential Equations

2011 ◽  
Author(s):  
Sun Yi ◽  
Sangseok Yu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Delay Systems ◽  
Lambert W Function ◽  
Short Paper ◽  
Characteristic Roots ◽  
General Systems ◽  
Delay Differential ◽  
Estimation Of Time

In this short paper, the preliminary result of a new method for estimation of time-delays of time-delay systems is presented. The presented method makes use of the Lambert W function, and is for scalar first-order delay differential equations (DDEs). Possible extension to general systems of DDEs and application to physical systems are also discussed.

Homotopy continuation for characteristic roots of delay differential equations using the Lambert W function

2017 ◽  
Vol 24 (17) ◽  
pp. 3944-3951 ◽  
Author(s):  
Samukham Surya ◽  
C. P. Vyasarayani ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Differential Equations ◽  
Characteristic Equation ◽  
Delay Differential Equations ◽  
Continuation Method ◽  
Lambert W Function ◽  
Homotopy Continuation ◽  
Multiple Delays ◽  
Homotopy Continuation Method ◽  
Characteristic Roots ◽  
Delay Differential

In this work, we develop a homotopy continuation method to find the characteristic roots of delay differential equations with multiple delays. We introduce a homotopy parameter μ into the characteristic equation in such a way that for μ = 0 this equation contains only one exponential term (corresponding to the largest delay) and for μ = 1 the original characteristic equation is recovered. For μ = 0, all the characteristic roots can be expressed in terms of the Lambert W function. Pseudo-arclength continuation is then used to trace the roots as a function of μ. We demonstrate the method on several test cases. Cases where it may fail are also discussed.

scholarly journals Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Guodong Liu ◽  
Xiaohong Wang ◽  
Xinzhu Meng ◽  
Shujing Gao
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Persistence In Mean ◽  
Predator Prey Model ◽  
Prey System ◽  
Delay Differential ◽  
Lévy Jumps

In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.

scholarly journals Numerical Stability Test of Neutral Delay Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Z. H. Wang
Keyword(s):  
Characteristic Function ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Characteristic Root ◽  
Initial Guess ◽  
Lambert W Function ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
The Stability ◽  
Delay Differential

The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

Pole Placement for Delay Differential Equations with Time-Periodic Delays Using Galerkin Approximations (CND-20-1378)

2021 ◽  
Author(s):  
Shanti Swaroop Kandala ◽  
Thomas K. Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Second Order ◽  
Delay Systems ◽  
First Order ◽  
The Galerkin Method ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems. In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory, and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

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