Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

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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“…Engelborghs and Doedel [23] have proven that the convergence rate of the maximal continuous error E = max t∈ [0,1] …”

Section: Collocationmentioning

confidence: 99%

Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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“…In addition, the stability of a class of collocation methods is analyzed in [21], convergence is shown to correspond to known convergence results for initial value delay equations in [22], and a technique for approximating connecting orbits for delay differential equations with stable or unstable manifolds of infinite dimension is developed in [36].…”

mentioning

confidence: 98%

“…The approximation of periodic solutions and connecting orbits of delay differential equations has been considered in [22,21,36] where the problems are formulated as boundary value problems and approximated with collocation methods. In addition, the stability of a class of collocation methods is analyzed in [21], convergence is shown to correspond to known convergence results for initial value delay equations in [22], and a technique for approximating connecting orbits for delay differential equations with stable or unstable manifolds of infinite dimension is developed in [36].…”

mentioning

confidence: 99%

Computation of Mixed Type Functional Differential Boundary Value Problems

Abell¹,

Elmer²,

Humphries³

et al. 2005

SIAM J. Appl. Dyn. Syst.

106

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Abstract. We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model and Frenkel-Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals.Key words. mixed type functional differential equations, boundary value problems, traveling waves, collocation AMS subject classifications. 65L10, 65L20, 35K57, 74N991. Introduction. Nonlinear spatially discrete diffusion equations occur as models in many areas of science and engineering. When the underlying mathematical models contain difference terms or delays as well as derivative terms, the resulting differential-difference equations present challenging analytical and computational problems. We demonstrate how functional differential boundary value problems with advances and delays arise from such models, and describe a general approach for the numerical computation of solutions. Solutions are approximated for several such problems, and the numerical issues arising in their computation are discussed.Biology, materials science, and solid state physics are three fields in which accurate first principle mathematical models possess difference (both delayed and advanced) terms. In biology (in particular, in physiology) there is the bidomain model for cardiac tissue (defibrillation), ionic conductance in motor nerves of vertebrates (saltatory conduction), tissue filtration, gas exchange in lungs, and calcium dynamics. Material science applications include interface motion in crystalline materials (crystal growth) and grain boundary movement in thin films where spatially discrete diffusion operators allow description of the material being modeled in terms of its underlying crystalline lattice. In solid state physics applications include dislocation in a crystal, adsorbate layers on a crystal surface, ionic conductors, glassy materials, charge density wave transport, chains of coupled Josephson junctions, and sliding friction. In all of these fields the physical system, and the corresponding differential model with delay terms, exhibit propagation failure (crystallographic pinning, a mobility threshold) and directional dependence (lattice anisotropy) in a "natural" way. These phenomena do not occur "naturally" in the models without difference terms commonly used for the above applications, and are often added to such local models in an 'ad hoc' manner. The reason discrete phenomena are modeled with continuous models is the lack of analytical techniques ...

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“…More details on the methods can be found in the articles [37,20,17,16,21,41] or in [15]. For details on applying these methods to bifurcation analysis of sd-DDEs see [36].…”

Section: Methodsmentioning

confidence: 99%

“…The numerical methods used are explained briefly in section 7, more details can be found in the papers [37,20,17,16,21,36,41] and in [15]. Its functionality is hidden by and used through layers 2 and 3.…”

Section: Structure Of Dde-biftoolmentioning

confidence: 99%

Численный Бифуркационный Анализ Математических Моделей С Запаздыванием По Времени С Использованием Пакета Программ DDE-BIFTOOL

Luzyanina

Sieber²,

Engelborghs³

et al. 2017

Math.Biol.Bioinf.

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Abstract. Mathematical modelling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of the life sciences, e.g., population dynamics, epidemiology, immunology, physiology, neural networks. The time delays in these models take into account a dependence of the present state of the modelled system on its past history. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period and so on. Due to an infinite-dimensional nature of DDEs, analytical studies of the corresponding mathematical models can only give limited results. Therefore, a numerical analysis is the major way to achieve both a qualitative and quantitative understanding of the model dynamics. A bifurcation analysis of a dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. The package DDE-BIFTOOL is the first general-purpose package for bifurcation analysis of DDEs. This package can be used to compute and analyze the local stability of steady-state (equilibria) and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. In this paper we describe the structure of DDE-BIFTOOL, numerical methods implemented in the package and we illustrate the use of the package using a certain DDE system.

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Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Cited by 32 publications

References 21 publications

Continuation and Bifurcation Analysis of Delay Differential Equations

Continuation and Bifurcation Analysis of Delay Differential Equations

Computation of Mixed Type Functional Differential Boundary Value Problems

Численный Бифуркационный Анализ Математических Моделей С Запаздыванием По Времени С Использованием Пакета Программ DDE-BIFTOOL

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