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Erschienen in:
The Twin Semigroup Approach Towards Periodic Neutral Delay Equations Kapitel lesen Erstes Kapitel lesen

2023 | OriginalPaper | Buchkapitel

Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach

verfasst von : Dimitri Breda

Erschienen in: Controlling Delayed Dynamics

Verlag: Springer International Publishing

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Abstract

Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the evolution family associated to linear problems. The focus is on local stability of either equilibria and periodic orbits as well as on generic nonautonomous systems, for either delay differential and renewal equations.

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Vorheriges Kapitel Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator Approach
Nächstes Kapitel Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications
Fußnoten
1
For a primer on Fréchet derivatives see Ambrosetti and Prodi (1995, Chap. 1).
 
2
For ease of presentation, we restrict to the case \(h\ge \tau \). For the case \(h<\tau \) we refer the interested reader to Breda et al. (2012), Breda and Liessi (2018) and Breda and Liessi (2020b) concerning, respectively, DDEs, REs and coupled equations. The case \(h<\tau \) is usually tackled by a piecewise version of the method for the case \(h\ge \tau \), introducing just technicalities but no conceptual novelties.
 
3
Uniqueness follows since we assume the relevant IVP to be well-posed.
 
4
The fixed point is actually unique for sufficiently large N (Breda et al. 2012, Sect. 3.4).
 
5
For relevant comments on the feature of spectral accuracy see Breda (2023).
 
6
The choice of 0 as starting time is not restrictive at all given that Lyapunov exponents concern the long-time behavior.
 
7
If interested in attractors of nonlinear problems, one usually linearizes around a reference (generic) trajectory.
 
8
Lower exponents come either as \(\liminf \) or as upper exponents of the adjoint system. If (15) is regular both upper and lower exponents coincide and exist as exact limit.
 
9
The latter, of course, is responsible for infinitely-many Lyapunov exponents as already mentioned, yet they all accumulate at \(-\infty \) Breda (2010).
 
10
Krauskopf and Sieber (2023, Sect. 4) is an example of application of Farmer (1982).
 
11
To note that, in any case, in (14) the operator V acts on polynomials.
 
12
The relevant theoretical background including Floquet theory, Poincaré sections and maps as well as the principle of linearized stability for periodic orbits has been developed only recently and can be found in Breda and Liessi (2020a).
 
13
The auxiliary codes clencurt, mybarint and mybarwei are also included, whose implementation follows Berrut and Trefethen (2004), Trefethen (2000).
 
14
A trivial exponent is always present due to linearization.
 
15
Available by the author upon request.
 
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Metadaten
Titel
Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach
verfasst von
Dimitri Breda
Copyright-Jahr
2023
Buch
Controlling Delayed Dynamics

Print ISBN: 978-3-031-00981-5

Electronic ISBN: 978-3-031-01129-0

Copyright-Jahr: 2023

DOI
https://doi.org/10.1007/978-3-031-01129-0_4

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