On Differentiability of Solutions with Respect to Parameters in State-Dependent Delay Equations

Hartung, Ferenc; Turi, János

doi:10.1006/jdeq.1996.3238

Cited by 67 publications

(69 citation statements)

References 6 publications

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“…The theory suggests that a Hopf bifurcation theorem holds; see, e.g., [57]. The stability of periodic solutions has only recently been studied; see, e.g., [40] for non-autonomous sd-DDEs. It was proven that the Fréchet derivative of the solution operator of the nonlinear sd-DDE with respect to initial data equals the solution operator of the linearized equation.…”

Section: State Dependent Equationsmentioning

confidence: 99%

Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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Mathematical modeling with delay differential equations (DDEs) is widely used in various application areas of science and engineering (e.g., in semiconductor lasers with delayed feedback, high-speed machining, communication networks, and control systems) and in the life sciences (e.g., in population dynamics, epidemiology, immunology, and physiology). Delay equations have an infinite-dimensional state space because their solution is unique only when an initial function is specified on a time interval of length equal to the largest delay. Consequently, analytical calculations are more difficult than for ordinary differential equations andnumerical methods are generally the only way to achieve a complete analysis, prediction and control of systems with time delays.Delay differential equations are a special type of functional differential equation (FDE). In FDEs the time evolution of the state variable can depend on the past in an arbitrary way as long as the dependence is a bounded function of the past. However, DDEs impose a constraint on this dependence, namely that the evolution depends only on certain past values of the state at discrete times. (We do not consider here the case of distributed delay.) The delays can be constant or state dependent. The equations can also involve delayed values of the derivative of the state, which leads to equations of neutral type.In this chapter we mainly discuss the simplest case, namely a finite number of constant delays. Specifically, we consider a nonlinear system of DDEs with constant delays τ j 0, j = 1, . . . , m, of the formwhere x(t) ∈ R n , and f : R (m+1)n+p → R n is a nonlinear smooth function depending on a number of (time-independent) parameters η ∈ R p . We assume that the delays are in increasing order and denote the maximal delay by

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Section: State Dependent Equationsmentioning

confidence: 99%

Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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“…При ε 3/2 опять получаем линейное неоднородное уравнение первого порядка с постоянным запаздыванием относительно функции u 3 : (10). Точное значение константы k нам не важно.…”

Section: построение нормальной формыunclassified

“…Но в прикладных задачах нередко возни-кают уравнения с запаздывающим аргументом, зависящим от искомой функции. В ряде работ установлено, что для анализа динамики и свойств таких уравнений во многих случаях применимы методы теории уравнений с постоянным запаздыванием (см., например, [7][8][9][10]). Вопросы существования и устойчивости решений рассмат-ривались, например, в [5,[11][12][13].…”

Section: Introductionunclassified

Local bifurcations analysis of a state-dependent delay differential equation

Golubenets

2016

Aut. Control Comp. Sci.

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“…Differentiability of the solutions of SD-DDEs wrt to the initial function (and other parameters) was studied in [10,19,21,22,24,35,36]. The only paper which proves differentiability of solutions of SDDDEs wrt the initial time σ is [21].…”

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confidence: 98%

Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays

Hartung

2015

J Dyn Diff Equat

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In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay. As an application, we discuss asymptotic properties of perturbed nonlinear differential equations with statedependent delays.

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On Differentiability of Solutions with Respect to Parameters in State-Dependent Delay Equations

Cited by 67 publications

References 6 publications

Continuation and Bifurcation Analysis of Delay Differential Equations

Continuation and Bifurcation Analysis of Delay Differential Equations

Local bifurcations analysis of a state-dependent delay differential equation

Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays

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