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)