Car following models are used to describe the behavior of a number of cars on the road dependent on the distance to the car in front. We introduce a system of ordinary differential equations and perform a theoretical and numerical analysis in order to find solutions that reflect various traffic situations. We present three different variations of the model motivated by reality

Gasser, Ingenuin

Seidel, Tilman

Werner, Bodo

Digitale Bibliothek Thüringen

17th International Conference on the Application of ComputerScience and Mathematics in Architecture and Civil EngineeringK. G¤urlebeck and C. K¤onke (eds.)Weimar, Germany, 1214 July 2006CAR FOLLOWING MODELS FOR PHENOMENA ON THE HIGHWAYT. Seidel , I. Gasser, and B. Werner Dep. Mathematik, Universit ¤at HamburgBundesstraße 55, 20146 HamburgE-mail: seidel@math.uni-hamburg.deKeywords: trafc, ow, models, stability, oscillationsAbstract. Car following models are used to describe the behavior of a number of cars on theroad dependent on the distance to the car in front.We introduce a system of ordinary differential equations and perform a theoretical and nu-merical analysis in order to nd solutions that reect various trafc situations. We present threedifferent variations of the model motivated by reality.11 INTRODUCTIONTrafc dynamics can be modeled with various mathematical techniques depending on thefocus the modeler sets. For example, from a macroscopic point of view, the cars are treated ascontinuum with a certain density. In contrast, microscopic models describe the movement ofeach individuel driver. Two surveys over the literature are [6, 8]. In this paper we concentrateon a class of microscopic trafc models.Consider a circular road of lengthwith  cars where 	 is the position of the  th driverat time  . The system of ordinary differential equationsﬀﬁﬂﬀﬃ "! #!ﬂ$%$%$ﬂ!&'! (1)describes the acceleration of each car depending on the headway ()*+ to the car in frontand its own velocity ﬃ, . With -.0/123*4the last car is always driving in front of therst one. Here  models the reaction time of the drivers and  is a so-called optimal velocityfunction of the headway.  is always positive, monotonically increasing and satises 56785 ,9;:=<?>"@BACDFE.G>(Fig. 1).System (1) is called a car following model and was rst mentioned by Bando [2].HIJ KLFEﬀG>MFigure 1: The general situation ( NPO'Q ) with an optimal velocity function R .For this type of trafc model we are interested in two different classes of solutions. Onthe one hand the free trafc where all cars drive with the same (constant) velocity around thecircle, and on the other hand the congested trafc. The second case may be observed as a kindof periodic solution where a trafc jam runs over the highway in the opposite direction to thedriving cars.How can we describe these two types of solutions mathematically?At rst we write (1) as a S -dimensional rst order systemTﬃ  UﬂﬃUﬂ V.WﬂﬀWUﬂﬂYX! Z#!ﬂ$%$%$ﬂ!&'$ (2)In a car following model, where the velocity of each driver depends only on the headway to thecar in front, a solution with constant velocity must also have constant headways. Therefore, wetransform (2) into a system in terms of the headways:Tﬃ[  \%ﬃ\ﬂ] [)^ﬀ_`["aW\%X! Z#!ﬂ$%$%$ﬂ!&'! (3)2where[b/1D)ﬀ+ is the headway between drivers b4and  , and \cd/1 ﬃ)ﬀeﬃ is thecorresponding relative speed. The headway[-. is equal to[ .Now it is easy to nd solutions with constant headways and all cars driving with the samevelocity. We are looking for stationary solutions of (3):T5  \%5  [)fﬀ_g[ﬂﬀW\ﬂﬁX! Z#!%$%$%$"!hY$A stationary solution [6i!&\i is given by[i\i 5!for each j!%$ﬂ$%$ﬂ!h . It corresponds to a so-called quasi-stationary solution i in terms ofsystem (1) withk7Dmlon34ﬁ! Z#!ﬂ$%$%$ﬂ!&'$Fig. 2 shows i for p!%$%$%$ !rq , a lengthDS5 , and a reaction time  .0 1 2 3 4 5 6048121620242832364044txiFigure 2: The quasi-stationary solution st with sktu%vxwzy O|{ w6}~ with {O|R  .Instead of simulations which are only able to nd stable solutions we are going to analysestationary solutions and their stability.2 STABILITYLet ﬀ[!)\F be the right hand side of system (3). The stability of the stationary solution[i!&\i is connected to the eigenvalues of the matrix/Ł80a[i!&\ir$3In [4] it is shown that the spectrum ofis given by the roots of the following equation:44++ﬂ6b! j!%$ﬂ$%$ﬂ!hY! (4)where -¡ . Consider that the eigenvalues as roots of Eq. (4) depend on two param-etersand  . Varying the number of cars  on the highway is a natural way, but in our casethis changes the dimension of the whole system. Therefore, we change the length of the roadand compute the eigenvalues ofunder variation of.Fig. 3 shows the eigenvalues lying in the Gaussian plane forDq and for5 . Althoughthere is always an eigenvalue 5 (compare (4) for `¢ ), it is shown in [4] that this eigenvaluedoes not inuence the stability of the quasi-stationary solution.−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2−1.5−1−0.500.511.5ℜℑ−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2−1.5−1−0.500.511.5ℜℑFigure 3: Eigenvalues of £¥¤ vx¦ tﬂ§z¨%t y for N©O|Q , ª«O|Q (left, unstable), and ª«O­¬® (right, stable).From Fig.3 we conclude that there is a region forwhere at least one pair of eigenvalues haspositive real parts and [i!&\i is unstable. Increasing, these eigenvalues cross the imaginaryaxis stabilizing the quasi-stationary solution (Fig. 4).−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2−1.5−1−0.500.511.5ℜℑFigure 4: Two eigenvalues crossing the imaginary axis under variation of ª«OYQ3¯f¯¯^¬® .In general an eigenvalue crossing the imaginary axis changes the stability of a stationarysolution. If there is pair of complex conjugated eigenvalues with zero real parts this leads4to Hopf bifurcations [7]. Under appropriate conditions, the theorem of Hopf guarantees theexistence of periodic solutionswhen stability changes take place under variation of a parameterlike. As an example, see Fig. 5 (left) which shows a periodic orbit in phase space togetherwith the unstable stationary point [°i!)\i .−0.5 0 0.5 1 1.5 2−1−0.8−0.6−0.4−0.200.20.40.60.81ξ1η 1−0.5 0 0.5 1 1.5 2 2.5−1−0.8−0.6−0.4−0.200.20.40.60.81ξ1η 1Figure 5: Two Phaseportraits ¨%± over ¦ ± : a stable stationary solution for ª²O©¬)® (right) and a stable periodicalsolution for ª#O'Q (left).These solutions being periodic in the headways of the cars and in the relative speeds, describea kind of congested trafc as in a stop-and-go situation. In fact, for trafc dynamics it is veryinteresting to show the existence of periodic solutions analytically. For an analysis of theirstability see [4].3 VARIATIONS OF THE MODELThere are many possibilities to vary system (1) to cope with different phenomena arising onthe highway. It is our goal to keep the model as simple as possible in order to obtain not onlynumerical but also theoretical results.3.1 Aggressive Driving BehaviorThe drivers in (1) change their velocity very 'moderately' dependent on an optimal velocityat a certain distance to the car in front. In reality everybody knows the situation driving on thehighway with an aggressive driver behind, blinking and changing his velocity immediately withoneself's velocity.This driving behavior can be modeled with an extra term in the model equation:Bc)ﬀ³ ´ﬃ74_µ·¶ﬃa´ﬃﬂ! p!%$%$%$ !h'$ (5)Here, µW¸P5 is an aggressive driving parameter. The more aggressive the driver the bigger is µ .In [5] we found out that the new µ -term does not change the (stability) analysis of the modelqualitatively  especially there are still quasi-stationary solutions [i!&\i and Hopf bifurca-tion points for a certain parameter constellation. Further more we showed, that for xedanincreasing µ can stabilize [i!&\i (see also [9]).53.2 Variable Reaction TimesOne major problem of (microscopic) car followingmodels is the possibility of negative head-ways which can be interpreted as a consequence of collisions in our model. In [5] we tried toreduce this risk by introducing a new reaction time    	F)a+  , depending on the head-ways as shown in Fig. 6: ).W  ).W ´ﬃ! #!%$%$%$ !h'$ (6)For the drivers on a highway the justication for this behavior is a more realistic modeling ofthe reaction for short distances  the closer the car in front, the better the concentration.ﬀWFigure 6: The variable reaction time ¹ v s uº ±» s u y .3.3 Road WorksMost of the car following models in literature have in common a model equation dependingon the headway independent of the certain position F of the  th driver. Our current interestfocusses on road works on the highway.Road works mathematically can be modeled by a maximum (optimal) velocity ¼EﬀG>thatnow depends on the position: there is a region on the circle, where the cars have to reduce theirspeed (Fig. 7).(E.G>½½(E.G>Figure 7: A region of reduced maximum velocity.The parameter ½ describes the 'size' of the road works which changes model (1) in the fol-lowing way:(E.G>c!½ﬀ¶¾	).W ﬀﬃﬂ¼$ (7)6For ½ ¿5 the maximum optimal velocity EﬀG>is constant as in (1), so (7) is a perturbation ofthe model without road works.Anyway, this new model is qualitatively different from the models in sections 3.1 and 3.2as the symmetry of the circle is broken. We can no longer hope to nd stationary or quasi-stationary solutions as each car changes its velocity and headway to the next car whenever itpasses the road works region  even for small ½ .A way out of this technical problem is to formulate a new (more general) solution type thatdescribes the movement of the cars on a circular highway with road works properly. In [1] thereis suggested a type of solutions called ponies on a merry-go-round (POM) that we adapted toour model.A POM solution is a solution that fulllls the following two conditions for a period À¢¸o5 .1. 4ÁÀ¡7Â	Ã4zÄÃ!ﬂ$%$%$ﬂ!&Å2. CÂ;ÆaÇ4ÉÈ-¡ zÄ.!ﬂ$%$%$"!hÅ .For this solution type it is possible (but mathematically more complex) to perform an analysisanalog to section 2.4 CONCLUSIONSThe car following models can help to understand the behavior of a nite number of cars onthe road from a microscopic point of view. The more drivers are involved into the system themore equations we need to describe their dynamics, increasing the complexity of numericalsimulations.The main mathematical tool is the analysis of stability and the use of bifurcation theory. Themodels can be understood easily in terms of one's own experiences on the highway.REFERENCES[1] Aronson D G, Golubitsky M, Mallet-Paret J, Ponies on a merry-go-round in large arraysof Josephson junctions. Nonlinearity 4 903-910, 1991.[2] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Dynamical model of trafccongestion and numerical simulation, Phys. Rev. E, vol. 51, p. 1035, 1995.[3] E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sand-stede, X. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differ-ential equation (with HomCont), Tech. Rep., Caltech, 2001.[4] I. Gasser, G. Sirito, B. Werner, Bifurcation analysis of a class of 'car following' trafcmodels, Physica D, vol. 197/3-4, pp. 222-241, 2004.[5] I. Gasser, T. Seidel, G. Sirito, B. Werner, Bifurcation analysis of a class of 'car following'trafc models II: variable reaction times and aggressive drivers, to appear in TransportTheory and Statistical Physics, 2005.7[6] D. Helbing, Trafc and related self-driven many-particle systems, Reviews of ModernPhysics, vol. 73, 2001.[7] E. Hopf, Abzweigung einer periodischen Lsung von einer stationa¨ren Lo¨sung eines Dif-ferentialsystems, Ber. Math.-Phys., S¤achsische Akademie der Wissenschaften Leipzig 94,1-22, 1942.[8] K. Nagel, P. Wagner, R. Woesler, Still owing: Approaches to trafc ow and trafc jammodeling, Operation Research, vol 51, No. 5, pp. 681-710, 2003.[9] L.E. Olmos, J.D. Munoz, A cellular automaton model for the trafc ow in Bogota, Int. J.Modern Physics C, 0406065, 2004.8

CAR FOLLOWING MODELS FOR PHENOMENONS ON THE HIGHWAY

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CAR FOLLOWING MODELS FOR PHENOMENONS ON THE HIGHWAY

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