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Multiscale Problems and Models in Traffic Flow, Vienna,
Austria, May 2008
Microscopic Models under a Macroscopic View
Ingenuin Gasser
Department of Mathematics
University of Hamburg, Germany
Tilman Seidel, Gabriele Sirito, Bodo Werner
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Outline:
• General
• Dynamics of the microscopic model (homogeneous case)
• Dynamics of the microscopic model (non-homogeneous case
and roadworks)
• Macroscopic view
• Fundamental diagrams
• Future
Basic concept: Take a very simple microscopic model (Bando),
study the full dynamics, take a macroscopic view on the results.
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Microscopic Bando model on a circular road (scaled)
N cars on a circular road of lenght L:
Behaviour: xj position of the j-th car
xj(t) = −
{
V(
xj+1(t) − xj(t))
− xj(t)}
, j = 1, ..., N, xN+1 = x1+L
V = V (x) optimal velocity function:
V (0) = 0, V strictly monoton increasing , limx→∞
V (x) = V max
0 1 2 3 4 5 6 7 8 9 10
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System for the headways: yj = xj+1 − xj
yj = zj
zj = −
{
V (yj+1) − V (yj) − zj
}
, j = 1, ..., N, yN+1 = y1
Additional condition:∑N
j=1 yj = L
“quasistationary” solutions: ys;j = LN
, zs;j = 0, j = 1, ..., N.
Linear stability-analysis around this solution gives for the Eigen-
values λ:
(λ2 + λ + β)N− βN = 0, β = V ′(
L
N)
Result (Huijberts (‘02)):
For 11+cos 2π
N
> βmax = maxxV ′(x) asymptotic stability
For 11+cos 2π
N
= V ′(LN
) loss of stability
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What kind of loss of stability? (I.G., G.Sirito, B. Werner ’04):
Eigenvalues as functions of β = V ′(LN
)
−2 −1.5 −1 −0.5 0 0.5 1−4
−3
−2
−1
0
1
2
3
4
Real Axis
Imag
inar
y ax
is
B
A O
A’
B’
C M
k=1
k=1
k=2 k=3 k=4
k=2
k=3 k=4
L/N
Bet
a
Bifurcation analysis gives a Hopfbifurcation.
Therefore we have locally periodic solutions.
Are these solutions stable? (i.e. is the bifurcation sub- or super-
critical?)
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Criterion: Sign of the first Ljapunov-coefficient l
Theorem:
l = c2
V ′′′
(
L
N
)
−
(
V ′′(LN
))2
V ′(LN
)
Conclusion: For the mostly used (Bando et al (95))
V (x) = V maxtanh(a(x − 1)) + tanh a
1 + tanh a
the bifurction is supercritical (i.e. stable periodic orbits).
But: “Similar” functions V give also subcritical bifurcations.
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Problem: It seems to be very sensitive with respect to V
Global bifurcation analysis: numerical tool (AUTO2000)
21.5 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
L
Nor
m o
f sol
utio
n
H1
21.5 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
L
Nor
m o
f sol
utio
n
H1
23.49 23.495 23.5 23.505 23.51 23.515 23.52 23.525 23.53
7.774
7.776
7.778
7.78
7.782
7.784
7.786
L
Nor
m o
f sol
utio
n
Conclusion: Globally “similar” functions V give similar behaviour.
The bifurcation is “macroscopically” subcritical
Conclusion for the application: the critical parameters from the
linear analysis are not relevant
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Example 1 Example 2 Example 3
10 12 14 16 18 20 22
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
L
Nor
m o
f sol
utio
n
H2
H1
21.5 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
L
Nor
m o
f sol
utio
n
H1
23.49 23.495 23.5 23.505 23.51 23.515 23.52 23.525 23.53
7.774
7.776
7.778
7.78
7.782
7.784
7.786
L
Nor
m o
f sol
utio
n
0.5 1 1.5 2 2.5 3 3.5−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Headway
Rel
ativ
e V
eloc
ity
0.5 1 1.5 2 2.5 3 3.5 4−1.5
−1
−0.5
0
0.5
1
1.5
Headway
Rel
ativ
e V
eloc
ity
1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Headway
Rel
ativ
e V
eloc
ity
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More bifurcations:
Eigenvalues as functions of β = V ′(LN
)
−2 −1.5 −1 −0.5 0 0.5 1−4
−3
−2
−1
0
1
2
3
4
Real Axis
Imag
inar
y ax
is
B
A O
A’
B’
C M
k=1
k=1
k=2 k=3 k=4
k=2
k=3 k=4
Conclusion: There are many other (weakly unstable) periodic
solutions
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(J.Greenberg ’04,’07) Solutions with many oscillations finally
tend to a solution with one oscillation
(G. Oroz, R.E. Wilson, B.Krauskopf ’04, ’05) Qualitatively the
same global bifurcation diagram for the model with delay
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Extension to “standart” microscopic model
Every driver is “aggressive” with weight α
xj(t) = −1 − α
τ
{
V(
xj+1(t) − xj(t))
− xj(t)}
+α{
xj+1(t) − xj(t)}
,
j = 1, ..., N, xN+1 = x1 + L optimal velocity-function:
loss of stability similar
“Aggressive” drivers stabilize the fraffic flow!
unfortunately also the number of accidents increases!
(Olmos & Munos, Condensed matter 2004)
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Road works
Vmax
ǫ
Symmetry breaking, the above theory is not easily applicable
A solution is called ponies on a Merry-Go-Round solution (short
POM), if there is a T ∈ R, such that
(i) xi(t + T) = xi(t) + L (i = 1, . . . , N)
(ii) xi(t) = xi−1
(
t + TN
)
(i = 1, . . . , N)
hold (Aronson, Golubitsky, Mallet-Paret ’91). We call T
rotation number and TN
the phase (phase shift).
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Theorem:The above model has POM solutions for small ǫ > 0.
0 2 4 6 8 10 12 14 16 18 203.9
3.92
3.94
3.96
3.98
4
4.02
0 2 4 6 8 10 12 14 16 18 203.725
3.73
3.735
3.74
3.745
3.75
3.755
3.76
3.765
3.77
x1
x1
Velocity of the quasistationary solution (no roadwork) versus
roadwork solution (The red line indicates maximum velocity).
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Technique: Poincare maps
Π(η) = ΦT(η)(η) − Λ, where Φ is the induced flow and Λ
reduces the spacial components by L.
b b
Σ Σ + Λx x + Λ
xǫ
ξ b
T
ΦT(ǫ,ξ)
−ΛΠ(ξ) bb
Study fixed points of the corresponding Poincare and reduced
Poincare maps. Roadworks are (regular) perturbations.
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Bifurcation diagram in (L, ǫ)-plane:
6 6.05 6.1 6.15 6.2 6.25 6.3 6.35 6.4 6.450
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
L
ε
I
II
A curve of Neimark-Sacker bifurcations in the (L, ǫ)-plane for
N = 5.
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Four different attractors.:
ǫ I II
ǫ = 0 trivial POM x0 Hopf periodic solution
ǫ > 0 POM xǫ quasi-POM
i.e. here
POM’s are typically perturbed quasistationary solutions
quasi-POM’s are perturbed (Hopf) periodic solutions
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Invariant curves:
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
headway 1/ρ
velo
city
v(ρ
)
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
headway 1/ρve
loci
ty v
(ρ)
Two closed invariant curves (ǫ = 0 and ǫ > 0) of the reduced
Poincare map π. On the left also the optimal velocity function
V0 is given in gray.
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The 4 different scenarios:
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1 mod L
v 1
L 10L0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
v 1
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1 mod L
v 1
L 10L0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
v 1
above: no roadworks, below: with roadworks
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Macroscopic view of the 4 different scenarios:
0 5 10 150
20
40
60
80
100
120
x mod L
t
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 2 4 6 8 10 120
20
40
60
80
100
120
x mod L
t
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 5 10 150
20
40
60
80
100
120
x mod L
t
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 2 4 6 8 10 120
20
40
60
80
100
120
x mod L
t
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
above: no roadworks, below: with roadworks
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Macroscopic view of density and velocity:
0 5 10 15 20 250
20
40
60
80
100
120
x mod L
t
1
2
3
4
5
6
7
0 5 10 15 20 250
20
40
60
80
100
120
x mod Lt
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
strong road work influence (ǫ = 0.32)
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Fundamental diagrams:
A real world point of view on the reduced Poincare map π for
N = 10, ǫ = 0.
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Fundamental diagrams I:
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ρ
ρ v(
ρ)
0 0.5 1 1.5 2 2.5 30.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ρ
ρ v(
ρ)
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
headway 1/ρ
velo
city
v(ρ
)
Overlapped fundamental diagrams for N = 10, L = 50, ...,4
measuring at a fixed point.
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Fundamental diagrams II:
0.5 1 1.5 2 2.5
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
N/L
ρ v(
ρ)
0.6 0.65 0.7 0.75 0.80.48
0.5
0.52
0.54
0.56
0.58
0.6
N/L
ρ v(
ρ)
Fundamental diagram of time-averaged flow versus average
density for N = 10, L = 50 . . .4.
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Fundamental diagrams III (with roadworks):
0 0.5 1 1.5 2 2.5 30.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
density ρi
traf
fic fl
ow ρ
i vi
Overlapped fundamental diagrams for
N = 10, L = 50, ...,4, ǫ = 0.1 measuring at a fixed point.
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Current and future work:
• Is this dynamics contained in macroscopic models?
• Which marcoscopic model has the same (rich) dynamics than
the basic Bando model
• Micro-macro link (Aw, Klar, Materne, Rascle 2002)