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

28 Chapter 2. Computing the spectrum
x
0.5
0
x
0.5
0
−0.5
−0.5
−1 −0.5
θ
0
(b) ϕ0(θ) = 0.5
−1 0 1
t
2 3
(a) The solution x(t). The dots • mark discontinuities in the derivatives
x
0.5
0
−0.5
−1 −0.5 0
θ
(c) T (τ)ϕ0
x
0.5
0
−0.5
−1 −0.5
θ
0
(d) T (2τ)ϕ0 =T (τ) 2 ϕ0
x
0.5
0
−0.5
−1 −0.5 0
θ
(e) T (3τ)ϕ0 =T (τ) 3 ϕ0
Figure 2.2: Graphical interpretation of Example 2.12 with initial condition ϕ0 =
0.5, τ = 1.
We now search for eigenvalues of the operator T (τ), i.e., µ ∈ C, and nontrivial
functions ϕ such that
µϕ = T (τ)ϕ.
Inserting the definition of T (2.15) for τ = h, yields
Moreover, we have,
µϕ ′ (θ) = − 3
ϕ(θ). (2.20)
2
µϕ(−τ) = ϕ(0). (2.21)
3 −
From (2.20), the eigenfunctions are ϕ(θ) = αe 2µ θ , where α is a normalization
constant. From the second condition (2.21) we have that ϕ(0) = α = µϕ(−τ) =
αµe3τ/2µ , i.e., µ =
with the corresponding eigenfunction ϕ(θ) =
−3τ
2Wk(−3τ/2)