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2023 | OriginalPaper | Buchkapitel

Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator 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 infinitesimal generator of the semigroup of solution operators associated to the system. The focus is on both local stability of equilibria and general bifurcation analysis of nonlinear problems, for either delay differential and renewal equations.

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Vorheriges Kapitel Characteristic Matrix Functions and Periodic Delay Equations

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

Recall that a compact linear operator \(K:X\rightarrow X\) satisfies \(\sigma (K)=\{0\}\cup \sigma _{p}(K)\) and either \(\sigma _{p}(K)\) has finitely-many points or they accumulate at 0.

We treat first DDEs for their longer tradition with respect to REs. Then, when dealing with coupled equations, we put REs first as in (1) following the convention in, e.g., Diekmann et al. (2008), due to the relative importance of REs with respect to DDEs in describing several models in population dynamics.

For a primer on Fréchet derivatives see (Ambrosetti and Prodi (1995), Chap. 1).

In what follows we use lower case letters for functions and upper case letters for their finite-dimensional counterparts, i.e., the vectors of their representation.

Anyway, let us also recall that Faber’s Theorem tells us that for any chosen set of nodes there is always a continuous function f for which \(\Vert p_{n}-f\Vert _{\infty }\) does not vanish, Faber (1914).

For an explanation of the term pseudospectral see (Brunner (2004), Sect. 1.7).

Note that pseudospacetral collocation is an alternative among many other ones, as, e.g., Galerkin-type or weighted-residuals approaches and series truncation. All these correspond in some sense to different choices of \(R_{n}\) and \(P_{n}\) and different strategies to obtain F. Note, however, that any choice must be suitably related to the features of X: e.g., there is no much sense in using Fourier series in \(C([a,b],\mathbb {R})\) or interpolation in \(L^{2}([a,b],\mathbb {R})\). Finally, let us remark that the choices above potentially lead to discretization approaches different from the one described in the rest of this chapter or in Breda (2023) when dealing with delay equations, see, e.g., Lehotzky and Insperger (2016) to name some examples.

For instance, if H acts on a domain \(\mathcal {D}(H)\subset X\) characterized by some given constraints (e.g., smoothness or boundary conditions), then the latter should be suitably taken into account in the discretization process, e.g., by explicitly replacing an equal number of collocation conditions (as, e.g., in Sect. 4.1) or by implicitly imposing such constraints in the construction of \(P_{n}\) (as, e.g., in Sect. 4.2).

Recall that in general a method of finite order, say m, applied to a function f of class \(C^{k}\) with \(k>m\) gives an error decaying as \(O(n^{-m})\) independently of k.

One can always rescale time to set \(\tau =1\) and hence fix the state space independently of the equation.

In fact, the range of \(P_{M}\) is not contained in \(\mathcal {D}(\mathcal {A}_{G})\), i.e., \((P_{M}\Psi )'(0)=L_{G}P_{M}\Psi \) does not hold for all \(\Psi \in Y_{M}\).

Let \(g\in C([-\tau ,0];\mathbb {R})\) and \(\{e_{1},\ldots ,e_{d_{Y}}\}\) be the canonical basis of \(\mathbb {R}^{d_{Y}}\). Then \(L_{G}g\) is a shorthand notation for the matrix \((L_{G}ge_{1}|\cdots |L_{G}ge_{d_{Y}})\in \mathbb {R}^{d_{Y}\times d_{Y}}\).

In fact, the range of \(P_{M}\) is now contained in \(\mathcal {D}(\mathcal {A}_{F})\) thanks to the first of (26).

A similar decompoistion is used below for \(L_{G}\) and note 12 applies to all \(L_{F,X}\), \(L_{F,Y}\), \(L_{G,Y}\) and \(L_{G,Y}\) with obvious modifications.

Indeed, Stirling’s formula for the factorial shows that the error decays as \(O(M^{-M})\).

Here “better” means that smaller values of M are required to reach a given tolerance.

Some work in this direction for DDEs can be found in Wu and Michiels (2012).

Although some indications are given in Breda et al. (2005), these may appear a little conservative and however a precise answer to the issue of choosing M is not available yet.

Here the term “abstract” refers to the infinite-dimensional state space on which the ODE is posed.

Recall in fact that \(P_{M}\) (and hence \(U_{0}\)) is defined only implicitly through the first of (31) (with F replacing \(L_{F}\)).

Just replace formally y(t) in (40) with \(e^{\lambda t}\).

Apart from the rightmost root \(\lambda =0\), we use as reference values for the other roots those computed with a very large value of M, viz. \(M=1000\). The same comment holds also for the other examples.

Concerning linerization of REs see (Diekmann et al. (2008), Sect. 3.5).

For a complete treatment see Breda et al. (2015a).

Formally, a coefficient \((\tau _{2}-\tau _{1})/2=1\) (for \(\tau _{1}=1\) and \(\tau _{2}=3\)) should appear in front of the sum as the weights are referred to the normalized integration interval \([-1,1]\).

It is left as an exercise to verify that this is indeed the rightmost couple (hint: write the characteristic equation and split it into real and imaginary parts).

For the smoothness of eigenfunctions of \(\mathcal {A}_{F}\) see (Breda and Liessi (2020), Sect. 5).

This value is approximated in Breda et al. (2013) by a pseudospectral approach slightly different from the one proposed here, difference due to an alternative treatment of the boundary condition in (10).

We did not report on the CPU time for the preceding examples as the computation of the eigenvalues is practically instantaneous.

Starting references to other techniques (and there are quite many of them nowadays) have been given in the introduction.

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Titel: Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator Approach
verfasst von: Dimitri Breda
Verlag: Springer International Publishing
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_3

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