Introduction to the Theory and Applications of Functional Differential Equations

Kolmanovskii, V. B.; Мышкис, А. Д.

doi:10.1007/978-94-017-1965-0

Cited by 1,160 publications

(862 citation statements)

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“…The precise value of the viscous friction is difficult to measure; however, we expect that it will be of a similar order of magnitude as β. Our work will thus consider in the range [0,15].…”

Section: X(t) Displacement Of the Center Of Mass Of The Cart From Poimentioning

confidence: 99%

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Dynamics of an Inverted Pendulum with Delayed Feedback Control

Landry¹,

Campbell²,

Morris³

et al. 2005

SIAM J. Appl. Dyn. Syst.

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Abstract. We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We describe a model for this system and use it to design a feedback control law that stabilizes the pendulum in the upright position. We then introduce a time delay into the feedback and prove that for values of the delay below a critical delay, the system remains stable. Using a center manifold reduction, we show that the system undergoes a supercritical Hopf bifurcation at the critical delay. Both the critical value of the delay and the stability of the limit cycle are verified experimentally. Our experimental data is illustrated with plots and videos.

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Section: X(t) Displacement Of the Center Of Mass Of The Cart From Poimentioning

confidence: 99%

“…Defining We now make use of some standard results from the stability theory of delay differential equations [5,15]. It was shown in section 3 that when τ = 0 all roots of (4.6) have negative real parts.…”

Section: Effect Of Time Delaymentioning

confidence: 99%

Dynamics of an Inverted Pendulum with Delayed Feedback Control

Landry¹,

Campbell²,

Morris³

et al. 2005

SIAM J. Appl. Dyn. Syst.

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“…Delay differential equations describe the models of phenomenons with delay effects such as stress-strain states of materials, population dynamics, and so on [9].…”

Section: Introductionmentioning

confidence: 99%

A Taylor operation method for solutions of generalized pantograph type delay differential equations

Yüzbaşı¹,

Ismailov²

2018

Turk J Math

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In this paper, a new operational matrix method based on the Taylor polynomials is presented to solve generalized pantograph type delay differential equations. The method is based on operational matrices of integration and product for Taylor polynomials. These matrices are obtained by using the best approximation of function by the Taylor polynomials. The advantage of the method is that the method does not require collocation points. By using the proposed method, the generalized pantograph equation problem is reduced to a system of linear algebraic equations. The solving of this system gives the coefficients of our solution. Numerical examples are given to demonstrate the accuracy of the technique and the numerical results show that the error of the present method is superior to that of other methods.The residual correction technique is used to improve the accuracy of the approximate solution and estimation of absolute error. The estimation aspect of the residual method is useful when the exact solution of the considered equation is unknown and this feature of the method is shown in numerical examples.

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“…Differential delay equations, or functional differential equations, have been used in modeling scientific phenomena for many years. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case it is called a distributed delay; see for instance the books by Hale and Verduyn Lunel [25], Kolmanovskii and Myshkis [30], Smith [40], and Wu [46], and the references therein. However, complicated situations in which the delay depends on the unknown functions have been considered in recent years.…”

Section: Introductionmentioning

confidence: 99%