The spectrum of delay-differential equations: numerical methods - KTH

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2.1. Introduction 11 For instance, we saw in the illustrative example in the previous chapter (Example 1.1) that the spectrum can be used to analyze stability, which is a property independent of the initial condition. For presentational reasons, it turns out to be advantageous to separate computational and numerical methods related to the spectrum into two contexts: • Approaches developed specifically for DDEs (Section 2.2) Even if we assume that the system is scalar, i.e., n = 1, the characteristic equation can not be solved for s using the standard exponential and trigonometric functions. There are however approaches exploiting properties of the DDE or the characteristic equation such that the problem can be accurately approximated or solved exactly for special cases. Two important concepts in this context are the two operators, the solution operator and the infinitesimal generator. There are methods specifically developed for the eigenvalues of DDEs derived from approximations of the DDE. Most methods are the result of approximations of these operators in ways which make the characteristic equation easier to solve. • Approaches stated in the more general context of nonlinear eigenvalue problems (Section 2.3) The problem of finding the eigenpairs of a DDE belongs to the class problems often called nonlinear eigenvalue problems. Here, a nonlinear eigenvalue problem is the problem of finding s ∈ C and v ∈ C n \{0} such that, T (s)v = 0, (2.3) for some parameter dependent matrix T : C → C n×n . The nonlinear eigenvalue problem is a very general class of problems, and there are no general methods which guarantee global convergence (to all solutions). There are however a number of methods with good local convergence properties. See [MV04] and [Ruh73] for overviews of numerical methods for nonlinear eigenvalue problems. The situation is somewhat better for some special cases, e.g. when the eigenvalues are real and T (s) is Hermitian. In particular, adapted projection methods have indeed been successfully applied to some nonlinear eigenvalue problems where T is rational and (2.3) has a (so-called) min-max characterization. See [VW82], [Wer72] and [Vos03].
12 Chapter 2. Computing the spectrum We present new results in both context. In particular, this chapter contains the following main contributions: • A clarification of the generality of an explicit formula containing the Lambert W function (Section 2.2.1). The spectrum of scalar single delay DDEs can be expressed explicitly using the inverse of the function ze z , referred to as the Lambert W function. One contribution of this chapter is the clarification of the range of applicability of this formula for the delay-eigenvalue problem. The expression is formed by defining a matrix version of Lambert W . The most general sufficient condition for which the formula holds is found to be simultaneously triangularizability of A0 and A1. • A unified interpretation for the methods based on discretization of the solution operator and the infinitesimal generator in terms of rational approximation (Section 2.2.2 and Section 2.2.3). We propose some new interpretations of current methods for the delay eigenvalue problem. In particular, we point out that several methods in the literature correspond to rational approximations or interpolations. This holds for instance for the methods based on solution operator discretizations (SOD). We point out that the SOD-method in the software package DDE-BIFTOOL is a companion linearization of a high-order rational eigenvalue problem. It turns out that this rational eigenvalue problem is a Padé approximation of the logarithm. We also discuss the methods based on discretizations of the PDE-formulation of the DDE, which appears to be superior for the numerical examples. • An application of a projection method to a delay eigenvalue problem (Section 2.3). The direct application of projection methods to the delay eigenvalue problems seems to be new. For this reason, it is worthwhile to clarify how projection methods can be adapted for this problem (Section 2.3.2). We adapt a subspace acceleration of the vector iteration, residual inverse iteration (RII) suggested by Neumaier in [Neu85]. This is an Arnoldi-type projection method and has been used by Voss in [Vos04a] for several other nonlinear eigenvalue problems, e.g., rational eigenvalue problems. In this projection method a small projected nonlinear eigenvalue problem must be
Page 1: The spectrum of delay-differential
Page 4 and 5: Acronyms DDE delay-differential equ
Page 6 and 7: 3.3.2 Plotting critical curves . .
Page 8 and 9: tion. The formula is not new. We cl
Page 10 and 11: Matrix-Version der Lambert W-Funkti
Page 12 and 13: Chapter 1 Introduction Many physica
Page 14 and 15: An illustrative example The main co
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2.4. Numerical examples 61 π/(n +
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2.4. Numerical examples 63 We made
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2.4. Numerical examples 65 2.4.2 Ra
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2.4. Numerical examples 67 ance 10
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70 Chapter 3. Critical Delays where
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72 Chapter 3. Critical Delays 3.1 I
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74 Chapter 3. Critical Delays For r
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116 Chapter 3. Critical Delays Lemm
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146 Chapter 4. Perturbation of nonl
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172 Appendix A. Appendix by ⎛ ⎜
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174 BIBLIOGRAPHY [Bau85] H. Baumgä
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176 BIBLIOGRAPHY [Bre06] , Solution
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178 BIBLIOGRAPHY [Dat78] R. Datko,
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180 BIBLIOGRAPHY [GMO + 06] M. Gu,
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182 BIBLIOGRAPHY [HS05] P. Hövel a
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184 BIBLIOGRAPHY [Lam91] J. Lambert
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186 BIBLIOGRAPHY [MGWN06] W. Michie
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188 BIBLIOGRAPHY [Nad69] S. B. Nadl
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190 BIBLIOGRAPHY [Pio07] M. Piotrow
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192 BIBLIOGRAPHY [SO06c] , A unique
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194 BIBLIOGRAPHY [Vos06a] , Iterati
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Index abstract Cauchy problem, 41,
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198 INDEX Nyquist criterion, 83 off
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Address: Institut Computational Mat
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The spectrum of delay-differential equations: numerical methods - KTH

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