Mathematical Models of Traffic Flow, Luminy, France, October 27 Microscopic Models under a Macroscopic Perspective 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: x j position of the j-th car ẍ j (t) = { V ( x j+ (t) x j (t) ) ẋ j (t) }, j =,..., N, x N+ = x +L V = V (x) optimal velocity function: V () =, V strictly monoton increasing, lim x V (x) = V max 2 3 4 5 6 7 8 9
System for the headways: y j = x j+ x j ẏ j = z j ż j = { V (y j+ ) V (y j ) ż j }, j =,..., N, yn+ = y Additional condition: N j= y j = L quasistationary solutions: y s;j = L N, z s;j =, j =,..., N. Linear stability-analysis around this solution gives for the Eigenvalues λ: (λ 2 + λ + β) N β N =, β = V ( L N ) Result (Huijberts ( 2)): For For +cos 2π N +cos 2π N > β max = max x V (x) asymptotic stability = V ( N L ) loss of stability
What kind of loss of stability? (I.G., G.Sirito, B. Werner 4): Eigenvalues as functions of β = V ( L N ) 4 3 2 k=4 k=3 k=2 B k= Imaginary axis M C A A O Beta 2 3 k= k=2 k=3 B k=4 4 2.5.5.5 Real Axis L/N Bifurcation analysis gives a Hopfbifurcation. Therefore we have locally periodic solutions. Are these solutions stable? (i.e. is the bifurcation sub- or supercritical?)
Criterion: Sign of the first Ljapunov-coefficient l Theorem: ( ( ) l = c 2 V L V ( L N )) 2 N V ( N L ) Conclusion: For the mostly used (Bando et al (95)) V (x) = V maxtanh(a(x )) + tanh a + 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 (AUTO2) 9 9.2 7.786 8 9 7.784 8.8 Norm of solution 7 6 5 H 3 H Norm of solution 8.6 8.4 8.2 Norm of solution 7.782 7.78 7.778 4 3 H 2 H 4 8 7.8 7.6 H 7.776 7.774 5 2 25 L 2.5 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 L 23.49 23.495 23.5 23.55 23.5 23.55 23.52 23.525 23.53 L 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 Example 2 Example 3 8 9.2 7.786 7.5 7 9 7.784 Norm of solution 6.5 6 5.5 5 4.5 4 H Norm of solution 8.8 8.6 8.4 8.2 8 7.8 H Norm of solution 7.782 7.78 7.778 7.776 3.5 7.774 H 2 7.6 2 4 6 8 2 22 L 2.5 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 L 23.49 23.495 23.5 23.55 23.5 23.55 23.52 23.525 23.53 L.5.5.8.6.4.3 Relative Velocity.4.2.2.4 Relative Velocity.5.5 Relative Velocity.2...2.6.8.3.4.5.5 2 2.5 3 3.5 Headway.5.5.5 2 2.5 3 3.5 4 Headway.5.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Headway
More bifurcations: Eigenvalues as functions of β = V ( L N ) 4 3 2 k=4 k=3 k=2 B k= Imaginary axis M C A A O 2 3 k= k=2 k=3 B k=4 4 2.5.5.5 Real Axis Conclusion: There are many other (weakly unstable) periodic solutions
(J.Greenberg 4, 7) Solutions with many oscillations finally tend to a solution with one oscillation (G. Oroz, R.E. Wilson, B.Krauskopf 4, 5) Qualitatively the same global bifurcation diagram for the model with delay
Extension to standart microscopic model Every driver is aggressive with weight α ẍ j (t) = α { ( V xj+ (t) x j (t) ) ẋ j (t) } +α { ẋ j+ (t) ẋ j (t) }, τ j =,..., N, x N+ = x + 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 24)
V max ǫ Road works 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) x i (t + T) = x i (t) + L (i =,..., N) (ii) x i (t) = x i ( t + T N ) (i =,..., N) hold (Aronson, Golubitsky, Mallet-Paret 9). We call T rotation number and N T the phase (phase shift).
Theorem:The above model has POM solutions for small ǫ >. 3.77 4.2 3.765 3.76 4 3.755 3.98 ẋ 3.75 3.745 3.96 3.74 3.94 3.735 3.73 3.92 3.725 2 4 6 8 2 4 6 8 2 3.9 x Velocity of the quasistationary solution (no roadwork) versus roadwork solution (The red line indicates maximum velocity).
Technique: Poincare maps Π(η) = Φ T(η) (η) Λ, where Φ is the induced flow and Λ reduces the spacial components by L. Π(ξ) Λ ξ x x + Λ Σ Φ T(ǫ,ξ) Σ + Λ T Study fixed points of the corresponding Poincare and reduced Poincare maps. Roadworks are (regular) perturbations. x ǫ
Bifurcation diagram in (L, ǫ)-plane:..9.8.7.6 I ε.5.4.3 II.2. 6 6.5 6. 6.5 6.2 6.25 6.3 6.35 6.4 6.45 L A curve of Neimark-Sacker bifurcations in the (L, ǫ)-plane for N = 5.
Four different attractors.: ǫ I II ǫ = trivial POM x Hopf periodic solution ǫ > POM x ǫ quasi-pom i.e. here POM s are typically perturbed quasistationary solutions quasi-pom s are perturbed (Hopf) periodic solutions
Invariant curves:.8.8 velocity v(ρ).6.4 velocity v(ρ).6.4.2.2.5.5 2 2.5 headway /ρ.5.5 2 2.5 headway /ρ Two closed invariant curves (ǫ = and ǫ > ) of the reduced Poincaré map π. On the left also the optimal velocity function V is given in gray.
The 4 different scenarios:.9.9.8.8.7.7.6.6 v.5 v.5.4.4.3.3.2.2.. 2 4 6 8 2 4 6 x mod L L x L.9.9.8.8.7.7.6.6 v.5 v.5.4.4.3.3.2.2.. 2 4 6 8 2 4 6 x mod L L x L above: no roadworks, below: with roadworks
Macroscopic view of the 4 different scenarios: 2 2 6 6 5.5 5.5 5 5 4.5 8 4.5 8 6 3.5 4 t t 4 6 3.5 3 4 3 4 2.5 2.5 2 2 2 2.5.5 5 5 2 4 x mod L 2 6 x mod L 8 2 2 6 6 5.5 5.5 5 5 4.5 8 4.5 8 6 3.5 4 t t 4 6 3.5 3 4 2.5 3 4 2.5 2 2.5 2 2.5 5 x mod L 5 2 4 6 x mod L 8 2 above: no roadworks, below: with roadworks
Macroscopic view of density and velocity: 2 7 2.9 6 8.8 5.7 6 4 4 3.6 t t 8 6.5.4 4.3 2 2 2.2 5 5 x mod L 2 25. 5 5 2 x mod L strong road work influence (ǫ =.32) 25
Fundamental diagrams: A real world point of view on the reduced Poincaré map π for N =, ǫ =.
Fundamental diagrams I: ρ v(ρ).9.8.7.6.5.4.3.2. 2 3 4 5 6 7 ρ ρ v(ρ).9.8.7.6.5.4.3.2.5.5 2 2.5 3 ρ velocity v(ρ).9.8.7.6.5.4.3.2. 2 3 4 5 headway /ρ Overlapped fundamental diagrams for N =, L = 5,..., 4 measuring at a fixed point.
Fundamental diagrams II: ρ v(ρ).65.6.55.5.45.4.35.3.25.2 ρ v(ρ).6.58.56.54.52.5.5.5 2 2.5 N/L.48.6.65.7.75.8 N/L Fundamental diagram of time-averaged flow versus average density for N =, L = 5...4.
Fundamental diagrams III (with roadworks):.9.8.7 traffic flow ρ i v i.6.5.4.3.2.5.5 2 2.5 3 density ρ i Overlapped fundamental diagrams for N =, L = 5,...,4, ǫ =. 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 22)