Pseudospectral Discretization of Nonlinear Delay Equations: New Prospects for Numerical Bifurcation Analysis

Breda, Dimitri; Diekmann, Odo; Gyllenberg, Mats; Scarabel, Francesca; Vermiglio, Rossana

doi:10.1137/15m1040931

Cited by 59 publications

(112 citation statements)

References 28 publications

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“…Two curves are shown in the left panel, obtained as explained next. The solid curve with circles is the result of the approach in [6], implemented in Matlab. The system of (11) and (12) is first reduced to a system of ODEs via a pseudospectral discretization.…”

Section: Remarkmentioning

confidence: 99%

“…The resolution of these external problems is often the computational bottleneck when continuation techniques are applied to compute equilibria or periodic solutions, analyze their stability and detect relevant bifurcations, which are all common targets. This is the case of the approach proposed very recently in [6]. Its underlying idea is to reduce the original model to a system of ODEs, but when it comes to apply standard continuation tools for ODEs to such system the role of the external problems in determining the computational cost emerges rather clearly.…”

mentioning

confidence: 99%

“…We describe Daphnia in Section 3, highlighting the external problems and the technicalities involved with it. In this work we concentrate on the continuation of its equilibria as a starting point, thus the rest of Section 3 focuses on how to approach this problem, either with the general method proposed in [6] or with more specific tools as proposed in [32]. In Section 4 we explain our internal approach in full detail.…”

mentioning

confidence: 99%

“…The pseudo-arclength continuation is independent of the chosen parameterization. Often the original problem (1) is described with an explicit parameter λ ∈ R and hence it is rather formulated as G(u, λ) = 0 (6) for G : R n × R → R n and u ∈ R n the true unknown vector. In this case in (1) we set v = (u, λ), we talk about natural parameterization and we look for the solution branch (u(λ), λ) or, basically, for u(λ).…”

mentioning

confidence: 99%

“…At the end we report on some computational results on the numerical continuation of its equilibria, obtained by using the technique recently proposed for general REs/DDEs [6], as well as by emulating algorithms already available for Daphnia-like models [32] (see also [10,12]).…”

mentioning

confidence: 99%

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Numerical continuation and delay equations: A novel approach for complex models of structured populations

Andò¹,

Breda²,

Scarabel³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.

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Section: Remarkmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical continuation and delay equations: A novel approach for complex models of structured populations

Andò¹,

Breda²,

Scarabel³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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A comparison of bivariate pseudospectral methods for nonlinear systems of steady nonsimilar boundary layer partial differential equations

Magagula

Motsa

Sibanda

2020

Comp and Math Methods

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In this work, three pseudospectral methods, namely bivariate spectral relaxation method, spectral local linearization, and bivariate spectral quasilinearization method are analyzed for steady nonlinear nonsimilar boundary layer partial differential equations. Their accuracy and general performance are discussed. A system of three nonlinear coupled partial differential equations that models an unsteady three‐dimensional MHD‐boundary‐layer flow due to an impulsive motion of a stretching surface is used to demonstrate the accuracy, convergence, and general performance of the three pseudospectral methods. The general comparison of the three pseudospectral methods is presented in graphical and tabular forms. Computational times of each method is presented in tabular form, showing the skin friction and Nusselt number.

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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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Pseudospectral Discretization of Nonlinear Delay Equations: New Prospects for Numerical Bifurcation Analysis

Cited by 59 publications

References 28 publications

Numerical continuation and delay equations: A novel approach for complex models of structured populations

Numerical continuation and delay equations: A novel approach for complex models of structured populations

A comparison of bivariate pseudospectral methods for nonlinear systems of steady nonsimilar boundary layer partial differential equations

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

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