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

4 Chapter 1. Introduction
x
5
0
−5
0 5 10 15
t
(a) α = 1.4, h = 1
x
5
0
−5
0 5 10 15
t
(b) α = 2, h = 1
Figure 1.1: The solution of the hot shower problem
To analyze the stability of (linear homogeneous) ODEs we typically look
for solutions with the ansatz x(t) = x0e st . For the DDE (1.1) this yields the
characteristic equation
0 = −s − αe −sh . (1.2)
If all solutions of (1.2), i.e., eigenvalues of (1.1), lie in the complex open left half
plane, then the system is stable, cf. Figure 1.2.
Similar to ODEs, the solutions of the characteristic equation of a DDE can be
used to analyze the stability. Several other properties can also be characterized
with the solutions of the characteristic equation, which is why we discuss ways
to numerically compute some solutions of the characteristic equation, also called
the eigenvalues of the DDE, for large-scale DDEs in Chapter 2.
Not even the scalar problem (1.2) can be solved explicitly using the normal elementary
functions. However, if we use the logarithmic type function Wk defined
as the inverse of z ↦→ ze z , z ∈ C we can express the solution of (1.2) explicitly,
s = 1
h Wk(−hα).
Here k ∈ Z is the branch index. This function (which is not so widely known) is
called Lambert W in the literature. Even though it is not an element of the set of
elementary functions, it is available in mathematical software and useful in some