Dynamical Phenomena induced by Bottleneck

Transcription

1 March 23, Sophia Antipolis, Workshop TRAM2 Dynamical Phenomena induced by Bottleneck Ingenuin Gasser Department of Mathematics University of Hamburg, Germany Tilman Seidel, Gabriele Sirito, Bodo Werner

2 Outline General Dynamics of the microscopic model (without bottleneck) Dynamics of the microscopic model (with bottleneck) Macroscopic view (with bottleneck) From micro to macro (without bottleneck) Fundamental diagrams (with and without bottleneck) Future References Basic concept: Take a very simple microscopic model (Bando), study the full dynamics, take a macroscopic view on the results.

3 Dynamics of the microscopic model (without bottleneck) 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

4 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

5 What kind of loss of stability? (I.G., G.Sirito, B. Werner 4): Eigenvalues as functions of β = V ( L N ) 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= 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?)

6 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 ))+tanha +tanha the bifurction is supercritical (i.e. stable periodic orbits). But: Similar functions V give also subcritical bifurcations.

7 Problem: It seems to be very sensitive with respect to V Global bifurcation analysis: numerical tool (AUTO2) Norm of solution Norm of solution Norm of solution H 7.8 H L L 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

8 Example Example 2 Example Norm of solution H Norm of solution H Norm of solution H L L L Relative Velocity Relative Velocity.5.5 Relative Velocity Headway Headway Headway

9 More bifurcations: Eigenvalues as functions of β = V ( L N ) k=4 k=3 k=2 B k= Imaginary axis M C A A O 2 3 k= k=2 k=3 B k= Real Axis Conclusion: There are many other (weakly unstable) periodic solutions

10 (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

11 Dynamics of the microscopic model (with bottleneck) V max 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) 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).

12 Theorem:The above model has POM solutions for small ǫ > ẋ x Velocity of the quasistationary solution (no bottleneck) versus bottleneck solution (The red line indicates maximum velocity).

13 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) x Σ Λ ξ Φ T(ǫ,ξ) x+λ Σ+Λ T Study fixed points of the corresponding Poincare and reduced Poincare maps. bottleneck are (regular) perturbations. x ǫ

14 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

15 Invariant curves:.8.8 velocity v(ρ).6.4 velocity v(ρ) headway /ρ 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.

16 The 4 different scenarios: v.5 v x mod L L x L v.5 v x mod L L x L above: no bottleneck, below: with bottleneck

17

18 Macroscopic view of the 4 different scenarios: t t x mod L 2 6 x mod L t t x mod L x mod L 8 2 above: no bottleneck, below: with bottleneck

19 From micro to macro (for simplicity without bottleneck) We start with Bando: x j ẍ j (t) = { ( V xj+ (t) x j (t) ) ẋ j (t) }, j =,...,N, x N+ = x +L τ ( ) xj+ (t)+x density: ρ j (t) 2,t = x j+ (t) x j (t) equilibrium velocity function: V e (ρ( x j++x j 2,t)) = V(x j+ x j ) asymptotics: small parameters (l, L are micro and macro lenght): ǫ = l L, τ position and velocity: x x j (t), zeroth order (in ǫ,τ): ρ t +(ρv e (ρ)) x = v(x,t) v j (t) = x j (t)

20 zeroth order (in ǫ) : ρ t +(ρv) x = (ρv) t + ( ρv 2) x = τ ρ(v e(ρ) v) first order (in ǫ) : (see also Aw, Rascle, Klar, Materne 3) ρ t +(ρv) x = ( (ρv) t + ρv 2 ǫ ) 2τ V e(ρ) = τ ρ(v e(ρ) v) x Question: are the Hopf periodic solutions in the microscopic world (appearing as traveling waves in the macroscopic presentation) traveling waves of macroscopic equations (scalar or system)? (see Greenberg,4, see Aw, Rascle, Klar, Materne 3, see Flynn, Kasimov, Nave, Rosales, Seibold 9)

21 Fundamental diagrams I: ρ v(ρ) ρ ρ v(ρ) ρ velocity v(ρ) headway /ρ Overlapped fundamental diagrams for N =,L = 5,...,4 measuring at a fixed point.

22 Fundamental diagrams II: ρ v(ρ) ρ v(ρ) N/L N/L Fundamental diagram of time-averaged flow versus average density for N =,L =

23 Fundamental diagrams III (with bottleneck): traffic flow ρ i v i density ρ i Overlapped fundamental diagrams for N =, L = 5,...,4, ǫ =. measuring at a fixed point.

24 How to open the ring road (B. Werner) (for simplicity without bottleneck) We start with Bando: x j ẍ j (t) = τ { V ( xj+ (t) x j (t) ) ẋ j (t) }, j =,...,N, x = x (t) is given the parameter L is somehow substituted by ẋ one can show easily: ẋ = const is an asymptotically stable solution of that system BUT: if N, the notion of stability has to be clarified! In fact: we encounter moving instabilities, i.e. the behaviour of every single car is stable, but there are perturbations which move backwards and increase! And there seems to be a bifurcation (similar to the ring road) separating the moving waves into stable and unstable ones.

25 Current and future work How to open the cicular road in the microscopic world? Is this dynamics contained in a macroscopic models? There is a time periodicity in the macroscopic pictures!

26 References I. Gasser, B. Werner: Dynamical phenomena induced by bottleneck, Philosophical Transactions of the Royal Society A, vol. 368 (928), (2). (Special Issue Highway Traffic - Dynamics and Control). I. Gasser, T. Seidel, B. Werner: Microscopic car-following models revisited: from road works to fundamental diagrams, SIAM J. Appl. Dyn. Sys., vol. 8, (29). I. Gasser, T. Seidel, G. Sirito, B. Werner: Bifurcation Analysis of a class of Car-Following Traffic Models II: Variable Reaction Times and Aggressive Drivers, Bulletin of Institute of Mathematics, Academia Sinica, New Series, vol. 2 (2), (27). I. Gasser, G. Sirito, B. Werner: Bifurcation analysis of a class of carfollowing traffic models, Physica D, vol. 97 (3-4), (24). I. Gasser: On Non-Entropy Solutions of Scalar Conservation Laws for Traffic Flow, Zeitschr. f. Angew. Math. Mech. ZAMM, vol. 83 (2), (23).