The spectrum of delay-differential equations: numerical methods - KTH
The spectrum of delay-differential equations: numerical methods - KTH
The spectrum of delay-differential equations: numerical methods - KTH
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Contents<br />
1 Introduction 1<br />
2 Computing the <strong>spectrum</strong> 9<br />
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
2.2 Methods for DDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />
2.2.1 Scalar or simultaneously triangularizable DDEs: Lambert W . . 16<br />
2.2.2 Solution operator discretization (SOD) . . . . . . . . . . . . . . . 24<br />
2.2.3 PDE discretization (IGD) . . . . . . . . . . . . . . . . . . . . . . 38<br />
2.3 Methods for nonlinear eigenvalue problems . . . . . . . . . . . . . . . . 46<br />
2.3.1 Scalar and vector valued fixed point <strong>methods</strong> . . . . . . . . . . . 47<br />
2.3.2 Projecton <strong>methods</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br />
2.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60<br />
2.4.1 A PDDE with real eigenvalues . . . . . . . . . . . . . . . . . . . 60<br />
2.4.2 Random matrix with complex eigenvalues . . . . . . . . . . . . . 65<br />
3 Critical Delays 69<br />
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
3.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
3.2 Notes and references on <strong>delay</strong>-dependent stability results . . . . . . . . . 82<br />
3.3 A parameterization for retarded DDEs . . . . . . . . . . . . . . . . . . . 90<br />
3.3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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