curvature of its spectrum in the origin. This property is already correctly represented in the lowest-order lowest-order macroscopic approximation and does not deteriorate for higher-lowest-order approximations.
Note that, unlike the microscopic case, the polynomial inνdiverges forν→ ±i∞and the essential spectrum will thus protrude arbitrarily wide into the right complex half plane in general . However, a realistic initial datum [r0, w0]>(ν) may be assumed to have bounded support, since wavelengths below the minimal distance of cars are not physically meaningful.
4.3. Examples
Example 4.2 (Index eigenvalues for different reference frames)
We again consider the simplistic first order traffic model vj = V(xj−1 −xj) with n = 1 and characteristic polynomialχ(λ, ξ) =λ+icξ−β(exp (iξ)−1). For c= 0, we have
heν =iξ = ln λ
β + 1
+ 2πil, l∈Z. On the other hand, for c6= 0 we have
heν =iξ=−λ+β
c −W
−β c exp
−λ+β c
(4.42) where W(z) denotes the multivariate Lambert W function of z ∈ C, returning the roots of the expression z−x·exp (x) (Fig. 4.4).
(a)c= 0 (b)c= 0.2 (c)c= 0.5
Figure 4.4: Spectra (top) and spatial eigenvalues for selected values ofλ(bottom) for the charac-teristic function χ =λ+icξ−β(exp (iξ)−1) of the first-order microscopic optimal velocity model. When changing to a co-moving coordinate system, the string stability properties stay the same but the roots ξ move
Example 4.3 (A Lyapunov function)
We demonstrate that the “weak coupling theorem” fromSwaroopand Hedrick(1996) is not a big help for the model in Ex. 4.1, since it is barely applicable for the affine-linearf∗:
By the triangle inequality,f∗ is globally Lipschitz with Lipschitz constantsl1,2 =β.
The origin of ˙x=f∗(x,0) is trivially globally exponentially stable with Lyapunov functionW(x) =
1
2x2, and the estimates
αlkxk2≤W(x)≤αhkxk2 (4.43a)
∂W
∂xf(x,0)≤ −α1kxk2 (4.43b)
∂W
∂x
≤α3kxk (4.43c)
are satisfied forαl=αh= 12,α1 =β,α3= 1.
As in the proof of the WCT in Swaroop and Hedrick (1996), we now pick 0 < d < 1 and sum the local Lyapunov functions W to the global Lyapunov function candidate W : RN → R,W(xj)j∈
N
=
∞
P
j=1
djW(xj). Global minimality ofW(0) is thus fulfilled. Analogously to the proof, we estimate ˙W:
d
dtW(xj)j∈J =
∞
X
j=1
dj ∂W
∂x x=x
j
f∗(xj, xj−1)
≤
∞
X
j=1
dj ∂W
∂x x=x
j
(f∗(xj,0) +l2kxj−1k)
≤
∞
X
j=1
dj−α1kxjk2+l2α3kxjk kxj−1k
≤
∞
X
j=1
dj
l2α3 2 −α1
kxjk2+ l2α3
2 kxj−1k2
≤0
Since l22α3 = 12α1 in this case, the last inequality is sharp, so it is not robust enough to make room for an additional nonlinearity.
The reason for this is obvious: since f∗ is antisymmetric in xj and xj−1, the coupling cannot be considered weak.
However, we may also confirm that W is a Lyapunov function by direct calculation (as pointed out by E. Felaco):
d
dtW(xj)j∈J =β
∞
X
j=1
xj(xj−1−xj) = −β 2
x21+
∞
X
j=1
x2j −2xjxj−1+x2j−1
= − β 2
x21+
∞
X
j=1
(xj−1−xj)2
<0
This approach has the advantage that it may be extended to (slightly) more realistic non-linear OVFsf: We first transfer the system to headway coordinateshj =xj−1−xj (note that, as before, this headway definition is shifted so that the QS corresponds to the origin), ˙hj =V(hj−1)−V(hj).
LetW(h) :=
Rh 0
V(y) dy and W(hj)j∈
N
:=
∞
P
j=1
W(hj).
Then we have d
dtW(hj)j∈J=
∞
X
j=1
V(hj) (V(hj−1)−V(hj)) =−1 2
∞
X
j=1
(V(hj−1)−V(hj))2.
Concavity ofW, induced by strict monotonicity ofV, is needed in order to show global minimality of W(0):
W(hj)j∈
N
=
∞
X
j=1
W(hj)> β
∞
X
j=1
hj = 0
4.3. Examples
can be written as
0 = det
0 1
−aβ −a
| {z }
A0
−λId + exp (iξ)
0 0
aβ 0
| {z }
A1
, (4.44)
whereβ=V0(he). Since n= 2, we are now dealing with block matrices and -operators (Fig. 4.5).
A0
A1 A0
A1 . ..
. ..
. ..
. .. A0
A1 A0
A1
(a) Circulant matrix
A0
A1 A0
A1 . ..
. ..
. ..
. .. A0
A1 A0
(b) Toeplitz matrix
A0
A1 A0
A1 A0
A1 . ..
. ..
(c) Toeplitz operator
A0
A1 A0
A1
A0
A1
. ..
. ..
. ..
. ..
(d) Laurent operator
Figure 4.5: Structure of relevant matrices and operators for the Bando model
The spectra of the circulant matrix, Laurent- and Toeplitz operator are given by (Fig. 4.6)
σ =
λ∈C: λ=−a 2±
s a2
4 +aβ(z−1), z∈
{z∈C:zN = 1} circulant matrix {z∈C:|z|= 1} Laurent operator {z∈C:|z| ≤1} Toeplitz operator
.
(4.45) The spectrum of the Toeplitz matrix does not depend onN and is given the eigenvalues ofA0,
σT M =
−a 2 ±
s a2
4 −aβ
. (4.46)
(a)N= 5 (b) N= 10 (c)N= 50 (d)N =∞
Figure 4.6: Spectra of the Bando model for parameters a = 1, he = 1.3. Note that the circular road model is stable for N = 5, but unstable for N ≥10
Example 4.5 (Pseudospectra and -modes of the Bando model) The pseudospectra
σε(AN) ={λ∈C:|det (AN −λId)| ≤ε} (4.47) for the Toeplitz matricesAN of the Bando model (Fig. 4.5(b)) can be calculated numerically with the package EIGTOOL for different values ofεand N (Fig. 4.7). We observe that for fixed ε >0, the pseudospectra appear to converge towards the spectrum of the Toeplitz operator as N → ∞.
(a)N = 10 (b)N= 50 (c)N = 100 (d)N= 500
log(ε)
Figure 4.7: ε-pseudospectra of the Toeplitz matrices for the Bando model, calculated with EIGTOOL. As N → ∞ and ε → 0, the ε-pseudospectra approximate the (essential) spectrum of the Toeplitz operator (cf. Fig. 4.6(d))
Example 4.6 (The characteristic polynomial of macroscopic models)
We again consider the Bando model with aggressive drivers (2.23) from Ex. 2.1. For brevity, we set a= ατ and γ = 1−ατ .
Locally, the linearisation is given by
y¨j = a(β(yj−1−yj)−y˙j) +γ( ˙yj−1−y˙j). (4.48) In Eulerian coordinates, the microscopic characteristic polynomial is
(λ+icξ)·(λ+icξ+a−γexp (iξ))−aβ(exp (iξ)−1). (4.49) Linearising the PDEs we derived in Ex. 3.2, we find the linearisation up tonth order inεfor the IHD model (3.39) is
r˜t+ver˜x = − 2ρ2e ε
bn/2c
X
j=0
w(2j+1) (2j+ 1)!
ε 2ρe
2j+1
(4.50a)
wt+vewx =aβ
− ε ρ2e
n−1
X
j=0
r(j) j!
ε 2ρe
j
−aw+γ
n
X
j=0
w(j) j!
ε ρe
j
−w
. (4.50b)
4.3. Examples
The corresponding characteristic polynomial is
(λ−νve)·
λ−νve+a−γ
n
X
j=0
1 j!
εν ρe
j
−2aβ
bn/2c
X
j=0
1 (2j+ 1)!
εν 2ρe
2j+1
·
n−1
X
j=0
1 j!
εν 2ρe
j
. (4.51) Linearisation of the forward-looking headway (3.40) yields
rˆt+ ˆrxve = −ρe2
ε
n+1
X
j=1
w(j) j!
ε ρe
j
(4.52a)
wt+vewx = −aβr ε
ρ2e +aw+γ
n
X
j=0
w(j) j!
ε ρe
j
−w
(4.52b)
with characteristic polynomial
(λ−νve)·
λ−νve+a−γ
n
X
j=0
1 j!
εν ρe
j
−aβ
n+1
X
j=1
1 j!
εν ρe
j
. (4.53)
Finally, for the natural density in (3.41) we obtain
rˇt+ ˇrxve = −ρewx (4.54a)
wt+vewx =aβ − ε ρ2e
n
X
k=1
r(k−1) k!
ε ρe
k−1!
−aw+γ
n
X
j=0
w(j) j!
ε ρe
j
−w
. (4.54b) Interestingly, this yields the same characteristic polynomial (4.51) as PDE (4.50). As n → ∞, both (4.51) and (4.53) converge towards the microscopic characteristic polynomial (4.49).
Chapter 5
Linear analysis of jam behaviour
In this chapter, we are investigating the question in which direction a small perturbation to an unstable quasistationary solution of a given car-following model moves while it is growing. In traffic flow, this corresponds to the question of where an emerging traffic jam moves. The answer will of course depend on the frame of reference we have in mind. Two points of view are especially important: The drivers’ perspective (or rather that of a theoretical driver at equilibrium velocity ve who is not affected by the perturbation) and the perspective from the side of the road. As in Sec. 4.1.2, we will refer to these as the index- and road frame or, as in fluid mechanics, as the Lagrangian and Eulerian point of view, respectively.
In a second step, it is of course interesting to obtain upper and lower bounds for the speed of information in order to determine the area in x, t-space to which the result of a localised perturbation is confined. Once we are able to detect whether a perturbation is moving up- or downstream in a given reference frame, this problem is solved, too: Finding the velocities of the fronts a traffic jam is equivalent to finding the coordinate systems in which these are at rest.
Without loss of generality, we may therefore concentrate on the two aforementioned systems in the following.
5.1 Historical overview
Historically, the speed of a perturbation has often been associated with group velocities. Briefly, this can be explained as follows: For a linear PDEut=f(u, ux, . . .) with initial profile
u0(x) =e−εx2 ·eik0x, k0∈R, ε >0 a Fourier transform gives
u0(k)∼e−
(k−k0)2
4ε .
If an exponential ansatz u=eλt+ikx yields the local approximation
λ(k) =λ0+iσ(k−k0) (5.1)
to the essential spectrum, we have
u(k, t)∼e−(k−k0)
2
4ε e(λ0+iσ(k−k0))·t whereσ = ∂ik∂λ
k=k0
denotes the group velocity.
After the inverse Fourier transform we obtain
u(x, t)∼eλ0t·e−ε(x+σt)2 ·eik0x.
From this, one might argue naively that the instability is of convective nature if for all unstable wavenumbers kthe corresponding group velocitiesσ are real and in a certain interval.
While this approximation may be applicable for σ ∈ R, it breaks down for σ ∈ C, i.e. in a dissipative medium. The problem here is not so much that the solution might explode if (Re (σ))2 −(Im (σ))2 < 0, but mainly the distortion of the pulse because of the e2iRe(σ)Im(σ)tx -term.
This issue was raised when after publication of special relativity in 1905, W. Wien objected that, for a refractive index of light smaller than one (which occurs for absorptive media), phase and group velocity may well be higher than the speed of light. In a talk titled “Ein Einwand gegen die Relativtheorie der Elektrodynamik und seine Beseitigung” in 1907, A. Sommerfeld discussed this issue and, on the basis of thought experiments, pointed out that neither the phase nor the group velocity are of relevance here but what he called the “signal velocity”, which can be calculated by means of complex integration. These ideas were further worked out inSommerfeld (1914), Brillouin (1914) and later translated to English and republished in Brillouin and Sommerfeld (1960) (cf. Pryce 1961). The focus of this work is on the “signal velocity”, the phrases “convective/absolute stability” are not yet used. The key idea in their approach is a sophisticated variation of curves along which complex integrals are evaluated.
Central ideas from this line of work were later applied to microscopic traffic flow models byWard and Wilson.
Briggsworked, apparently unaware of Brillouin/Sommerfeld, on the interaction between electron-streams and plasmas. The key phrase here is the distinction between “temporal” and “spatial”
instability. The work is based on early ideas fromLifshitz; vice versa,Lifshitzand Pitaevskii (1981) refer closely to formalism in Briggs (1964). Brevdo (1988) can also be attributed to this line of thought, with interesting applications to periodic solutions later on in Brevdo and Bridges(1996, 1997) (to be discussed in Sec. 7.2).
Sherratt et al. (2014) give an introduction and overview to the topic from a biological point of view, with special emphasis on the ideas of Sandstedeand Scheelwho introduced the notions of transient and remnant instability which we will address in Sec. 5.3.
For traffic flow models, the topic was discussed by different authors:
– Treiber and Kesting (2011) use an approximation by group velocities and find good agreement with simulations
– Mitarai and Nakanishi (1999, 2000a,b) use the PDE formalism from Lifshitz and Pitaevskii(1981) for the microscopic Bando model