View
221
Download
2
Category
Preview:
Citation preview
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
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.
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
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
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?)
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.
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
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
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
(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
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)
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).
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).
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.
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.
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
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.
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
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
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)
Fundamental diagrams:
A real world point of view on the reduced Poincare map π for
N = 10, ǫ = 0.
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.
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.
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.
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)
Recommended