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

10 Chapter 2. Computing the spectrum
the characteristic equation of (2.1);
ii) a solution s ∈ C to the characteristic equation (2.2) an eigenvalue;
iii) the set of all solutions of (2.2) the spectrum of the DDE (2.1) and denote
it by
σ(Σ) := � s ∈ C : det � −sI + A0 + A1e −τs� = 0 � ;
iv) a vector v ∈ C n \{0} corresponding to the eigenvalue s ∈ σ(Σ), an eigenvector
corresponding to s iff
� −sI + A0 + A1e −τs� v = 0,
and the pair (s, v) is called an eigenpair.
Moreover, the term delay eigenvalue problem (DEP), will be used to refer to the
problem of finding eigenpairs of a DDE.
This is a true generalization of the eigenvalues and the characteristic equation
of a matrix since if τ = 0 or A1 = 0 then the DDE reduces to an ODE and
(2.2) coincides with the characteristic equation of a matrix. Unlike the delayfree
case, the characteristic equation generally contains an exponential term.
This exponential term (and more generally the characteristic equation) is, in
the literature, commonly motivated by looking for non-trivial solutions using the
exponential ansatz x(t) = e st v. If we insert the exponential ansatz into (2.1) we
arrive at the characteristic equation. See e.g. [HV93] or [MN07b] for more formal
settings.
Note that, the use of the term spectrum in Definition 2.1 is consistent with
the common use of the term. The spectrum is normally a property of an operator
or a matrix. The terminology of eigenvalues and spectrum of a DDE is indeed
consistent, since the DDE (2.1) can be stated as a linear operator acting on a
Banach space which spectrum is a point-spectrum [HV93]. That is, the continuous
part and the residual part of the spectrum of the operator are empty, which
implies that the spectrum consists only of the set of eigenvalues of the operator,
i.e., consistent with Definition 2.1.
Also note that the spectrum is independent of the initial condition ϕ. Even
though the spectrum can be used in many different settings, it is particularly
often used to analyze properties which are independent of the initial condition.