A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

Lehotzky, Dávid; Insperger, Tamás

doi:10.1002/nme.5225

Cited by 18 publications

(9 citation statements)

References 39 publications

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“…On the other hand, computational costs can be reduced further by replacing semidiscretization with more effective discretization techniques. () This changes Equation only, whereas the subsequent analysis remains the same. These techniques operate with significantly smaller coefficient matrices and less matrix multiplications.…”

Section: Discussionmentioning

confidence: 99%

“…Several techniques exist for discretizing DDEs (see the works of Krauskopf et al, Loiseau et al, and Breda et al where some relevant approaches are collected). The most popular and most efficient numerical methods include the pseudospectral collocation,() the Chebyshev spectral continuous‐time approximation, the spectral element method, the spectral Legendre tau method, and the pseudospectral tau method . In what follows, we use the semidiscretization technique to discretize the solution operator of the DDE.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Molnár

Dombóvári

Insperger

et al. 2018

Numerical Meth Engineering

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Bifurcations of the periodic stationary solutions of nonlinear time-periodic time-delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay-differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map that describes the evolution over the time period. Fold, flip, and Neimark-Sacker bifurcations related to the fixed point of this map are analyzed via center manifold reduction and normal form theorems. The analysis unfolds the approximate stability properties and bifurcations of the stationary solution of the delay-differential equation and, at the same time, allows the approximate computation of the arising period-1, period-2, and quasi-periodic solution branches. The method is demonstrated for the delayed Mathieu-Duffing equation, and the results are verified by numerical continuation. 57 58 MOLNAR ET AL.the nonlinear time-periodic DDE using a nonlinear map that describes the evolution over the time period. The fixed point of this map corresponds to the stationary solution of the original DDE. Fold, flip, and Neimark-Sacker bifurcations of the fixed point are associated with cyclic fold, period doubling, and secondary Hopf bifurcations of the stationary solution, respectively, which may give rise to period-1, period-2, and quasi-periodic solutions. We determine the stability of these solutions by analyzing the fixed point of the corresponding nonlinear map and calculating its critical normal form coefficients. Via these coefficients, we use analytical formulas to obtain the approximate amplitude of the bifurcating solutions as a function of the bifurcation parameter. Therefore, as opposed to Knut, this method does not require the point-by-point continuation of the arising solutions; however, the results are accurate in the vicinity of the bifurcation point only, and secondary bifurcations cannot be detected.The first step of the analysis is to discretize the solution operator of the nonlinear time-periodic DDE. Several techniques exist for discretizing DDEs (see the works of Krauskopf et al, 9 Loiseau et al, 10 and Breda et al 11 where some relevant approaches are collected). The most popular and most efficient numerical methods include the pseudospectral collocation, 12,13 the Chebyshev spectral continuous-time approximation, 14 the spectral element method, 15 the spectral Legendre tau method, 16 and the pseudospectral tau method. 17 In what follows, we use the semidiscretization technique 18 to discretize the solution operator of the DDE. This method formulates a sequence of nonlinear maps that approximate the dynamics over the time period. Note that the approach of this paper is not restricted to semidiscretization; it supports other discretization techniques as well, as long as the solution operator is approximated by a (sequence of) nonlinear map(s) over the time period.We show an algorithm to build a single resultant map from the sequence of nonlinear maps that i...

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Molnár

Dombóvári

Insperger

et al. 2018

Numerical Meth Engineering

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“…This section details the application of the PT [23] and SE [21] methods for the approximation of the monodromy operator of the hybrid system (1). While the PT method discretizes the operator differential equation form, the SE method is based on the operator equation form of (1).…”

Section: Numerical Stability Analysismentioning

confidence: 99%

“…In this section, the main steps of the approximation schemes are given for the hybrid system (1). The full description of the PT and SE methods for continuous-time (non-hybrid) systems can be found in [23] and [21], respectively.…”

Section: Numerical Stability Analysismentioning

confidence: 99%

Numerical methods for the stability of time-periodic hybrid time-delay systems with applications

Lehotzky

Insperger

Stépàn

2018

Applied Mathematical Modelling

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This paper extends two numerical methods for the stability analysis of a class of time-periodic hybrid time-delay systems. In particular, the pseudospectral tau and spectral element methods are extended to hybrid systems. The analyzed delay-differential equation involves delayed terms with both continuous and piecewise constant arguments, in other words, it involves delays both without and with zero-order hold, respectively. The analyzed class of hybrid systems can be used to describe time-periodic hereditary processes subjected to digital feedback control. The proposed numerical algorithms are applied to the mathematical models of haptic systems and milling processes subjected to digital feedback control.

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“…Alternatively to frequency-domain methods, time-domain methods-such as the semidiscretization method (SDM [12][13][14]), full-discretization [15], temporal finite element analysis (TFEA, [16]), pseudospectral collocation [17], or pseudospectral tau [18] methods-can be applied, among others. Time-domain methods provide the Floquet multipliers [19] for combinations of the spindle speed and the depth of cut over a given or fractal mesh [20].…”

Section: Introductionmentioning

confidence: 99%

The Basics of Time-Domain-Based Milling Stability Prediction Using Frequency Response Function

Dombóvári

Sanz-Calle

Zatarain

2020

JMMP

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This study presents the fundamentals of the usage of frequency response functions (FRF) directly in time-domain-based methods. The methodology intends to combine the advantages of frequency- and time-domain-based techniques to determine the stability of stationary solutions of a given milling process. This is achieved by applying the so-called impulse dynamic subspace (IDS) method, with which the impulse response function (IRF) can be disassembled to separated singular IRFs that form the basis of the used transformation. Knowing the IDS state, the linear stability boundaries can be constructed and a measure of stability can be determined using the Floquet multipliers via the semidiscretization method (SDM). This step has a huge importance in parameter optimization where the multipliers can be used as objective functions, which is hardly achievable using frequency-domain-based methods. Here we present the basic idea of utilizing the IDS method and analyze its convergence properties.

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A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

Cited by 18 publications

References 39 publications

Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Numerical methods for the stability of time-periodic hybrid time-delay systems with applications

The Basics of Time-Domain-Based Milling Stability Prediction Using Frequency Response Function

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