Nonlinear dynamics of traffic jams

Physica D207(2005)41–51Nonlinear dynamics of traffic jamsTong Li∗Department of Mathematics,University of Iowa,14MacLean Hall,Iowa City,IA52242,USAReceived2September2004;received in revised form11May2005;accepted13May2005Available online6June2005Communicated by J.LegaAbstractA class of trafficflow models that capture the nonlinear dynamics of traffic jams are proposed.The class of discrete models originate from equations satisfied by the discrete traveling waves of some inviscid nonequilibrium continuum models.The self-organized oscillatory behavior and chaotic behavior in traffic are identified and formulated.The results can help to explain the appearance of a phantom traffic jam observed in real trafficflow.There is a qualitative agreement when the analytical results are compared with the empiricalfindings for freeway traffic and with the previous numerical simulations.©2005Elsevier B.V.All rights reserved.AMS Classification:35B30;35B40;35L65;76L05;90B20Keywords:Traffic jams;Chaos;Stop-and-go waves;Bifurcation;Discrete traveling waves;Nonconcavity1.IntroductionExperimental observations of real traffic have revealed rich nonlinear phenomena:the formation of traffic jams,stop-and-go waves,hysteresis and phase transitions,see Daganzo et al.[9],Helbing[19], Kerner[24,25],Kerner and Rehborn[27],Knospe, Santen,Schadschneider and Schreckenberg[29], Mauch and Cassidy[42],Treiterer and Myers[52]. Traffic systems exhibit extremely complex behavior derived from several sources:the heterogeneous nature of human behavior,highly nonlinear group Tel.:+13193343342;fax:+13193350627.E-mail address:tli@.dynamics and large system dimensions.Many mod-elling approaches have been suggested by traffic engineers,physicists and mathematicians.They use either discrete or continuous state space,with discrete or continuous time and/or space,see Bando et al.[2], Bellomo,Delitala and Coscia[3],Berg and Woods[4], Colombo and Groli[7],Gazis,Herman,and Rothery [11],Gray and Griffeath[12],Greenberg,Klar and Rascle[15],G¨u nther,Klar,Materne and Wegener[17], Hayakawa and Nakanishi[18],Helbing[20],Hermann and Kerner[21],Illner,Klar and Materne[22],Jin and Zhang[23],Kerner and Konh¨a user[26],K¨u hne [30],Li[36],Lighthill and Whitham[40],Nagel[43], Payne[45],Prigogine and Herman[46],Richards[47], Smilowitz and Daganzo[50],Zhang[55].Theoretical0167-2789/$–see front matter©2005Elsevier B.V.All rights reserved. doi:10.1016/j.physd.2005.05.01142T.Li/Physica D207(2005)41–51modelling allows simulations of large traffic networks and opens perspectives for future applications such as traffic forecasts and dynamic route guidance systems.Intrigued by the phenomenology of real-world traffic and the simulations of others,we propose an innovative approach to the nonlinear dynamics of trafficflow.The class of discrete models are derived from nonequilibrium continuum models.We are able to identify and predict the self-organized oscillatory paring our analytical results with the em-piricalfindings for freeway traffic,we see a qualitative agreement.Previously,Helbing[19]reviewed results on em-piricalfindings and modeling approaches of congested traffic.Gray and Griffeath[12]studied the ergodic theory of traffic jams.They analyzed a probabilistic cel-lular automaton(CA)as a prototype for the emergence of traffic jams at some intermediate density range,also see Nagel[43].Schadschneider and Schreckenberg [49]obtained stop-and-go waves from CA models.Kerner and Konh¨a user[26],Jin and Zhang[23] obtained clustering solutions when they numerically solved the viscous and inviscid Payne-Whitham Eqs.(9)and(10)respectively,with nonconcave fundamen-tal diagram(16)in the unstable regions.Clusters in trafficflow are characterized by the appearance of a region of high density and low average velocity of vehicles in an initially homogeneousflow.In the stable region(12)where the sub-characteristic condition is satisfied,Li and Liu[38]showed that the travel-ing wave solutions of the inviscid Payne-Whitham equations with nonconcave fundamental diagrams are asymptotically stable under small disturbances and under the sub-characteristic condition.In their numer-ical simulations of the viscous Payne-Whitham Eqs.(9)and(10)with a nonconcave fundamental diagram (18),Lee et al.[33]triggered a form of stop-and-go traffic.Greenberg,Klar and Rascle[15]found wave train solutions for a model for traffic on a multilane freeway.Bando et al.[2]proposed an optimal velocity model and observed the evolution of traffic congestion. Gasser,Sirito and Werner[10]performed bifurcation analysis of a class of car-following traffic models. Numerical simulations of Greenberg[14]yielded the periodic traveling wave solutions for a higher-order trafficflow model on a ring road with fundamental diagram(19).In all the continuum models,noncon-cave fundamental diagrams were used to obtain the oscillatory solutions.A key physical condition in obtaining clustering solutions in continuum models is that the fundamental diagrams change concavity.The paper is organized as follows.We present important approaches toward the modeling of traffic phenomena in Section2.We derive the discrete models in Section3.In Section4,we show existence of clustering solutions and compare them with the previous observations and numerical simulations.We conclude in Section5.2.PreliminariesThere are various important approaches to the mod-eling of traffic phenomena:microscopic models which explain traffic phenomena on the basis of the behavior of single vehicles[11],mesoscopic models such as kinetic or Boltzmann-like models[19,28,46],and macroscopic models which describe traffic phenom-ena through parameters which characterize collective traffic properties[1,8,13,35,36,39,40,45,53,55].Assuming that there exists a function relation be-tween the velocity and the density v=v e(ρ),Lighthill and Whitham[40]and Richards[47]developed the first macroscopic model of trafficflow,LWR(Lighthill, Whitham and Richards)theory,ρt+(ρv e(ρ))x=0.(1) The initial data isρ(x,0)=ρ0(x)>0.(2) v e(ρ)is a non-increasing functionv e(ρ)≤0.(3) v e(0)=v f and v e(ρj)=0,where v f is the freeflow speed andρj is the jam concentration.q(ρ)=ρv e(ρ)(4) is the so-called fundamental diagram in trafficflow. There are a huge number of different suggestions about the speed-density relation.The fundamental diagrams may be concave,see Greenshields[16],nonconcave, smooth,discontinuous or have multiple branches.We focus our attention on trafficflow models with single-valued fundamental diagrams that change concavity.ItT.Li/Physica D207(2005)41–5143 will be shown that a concavity change in the funda-mental diagram is a necessary and sufficient conditionin obtaining the oscillatory solutions.Experiment datasupports the condition that the fundamental diagramchanges concavity,see Daganzo et al.[9],Helbing etal.[20],Kerner and Rehborn[27].Eq.(1)is a nonlinear conservation law.It can ex-plain the formation of shock waves which correspondto congestion formation in trafficflow.The character-istic speed of(1)isλ∗(ρ)=q (ρ)=ρv e(ρ)+v e(ρ).(5)λ∗(ρ)is not faster than the traffic speed v e(ρ)underthe assumption that v e(ρ)≤0.This is the so-calledanisotropic property.A discontinuityρ(x,t)=ρl x≤stρr x>st(6)is a weak solution of(1)if the Rankine-Hugoniot con-ditions=q(ρl)−q(ρr)ρl−ρr(7)is satisfied.We say that the discontinuity defined in(6) and(7)satisfies the entropy condition ifq(ρr)−q(ρl)ρr−ρl ≤q(ρ)−q(ρl)ρ−ρl(8)for allρbetweenρr andρl.Under the entropy condition(8),Oleinik[44] proved the uniqueness of the weak solution of(1) and(2).The LWR theory fails in describing more compli-cated trafficflow patterns including hysteresis phenom-ena and stop-and-go traffic.This is due to the unrealistic assumption that the equilibrium speed is adapted in-stantaneously.The LWR model(1)has been extended to nonequilibrium macroscopic models that include the dynamics of the velocity[1,26,30,37,45,53,55].Kerner and Konh¨a user[26],K¨u hne[30]proposed the viscous nonequilibrium PW modelρt+(ρv)x=0(9)v t+vv x+c20ρx=v e(ρ)−v+µv xx(10)whereτ>0is the relaxation time,c0is the traffic soundspeed andµ>0is the viscosity coefficient.Eq.(9)is a conservation law forρ.Eq.(10)de-scribes drivers’acceleration behavior.The accelerationconsists of a relaxation to the static equilibrium speed-density relation and an anticipation which expressesthe effect of drivers reacting to conditions downstream.The third term on the left hand side of(10)accountsfor the anticipation effect.Its physical meaning is thatone tends to reduce speed when the density increases.The right hand side of(10)is the relaxation term andthe dissipative term.Whenµ=0,the inviscid model(9)and(10)is thePayne[45]and Whitham[53]model or PW model,which is a system of nonlinear hyperbolic equations.Its characteristic speeds areλ1=v−c0<λ2=v+c0.The model(9)and(10)is stable in the sense ofWhitham[53]ifλ1<λ∗<λ2(11)or−c0<ρv e(ρ)<c0(12)on the equilibrium curve v=v e(ρ).Condition(11)isthe so-called strict subcharacteristic condition.Underthe strict subcharacteristic condition(11),Liu[41]derived the nonlinear stability of elementary waves.Indeed,under the assumption(11),it was derived,inthe same spirit as the Chapman-Enskog expansion,that the relaxation process is approximated accuratelyby a viscous conservation lawu t+(f∗(u))x=(β(u)u x)x(13)whereβ(u)=−τ(u)(λ∗−λ1)(λ∗−λ2)>0.Assum-ing that the fundamental diagram q(ρ)is concave,Schochet[48]established the well-posedness forcertain inviscid Payne type higher order trafficflowttanzio and Marcati[32]showed the con-vergence to the equilibrium solution as the relaxationparameterτtends to zero.If the fundamental diagram q(ρ)is nonconcave,thenthere is an inflection pointρi.Thus there exists an un-stable region[ρc1,ρc2](14)44T.Li /Physica D 207(2005)41–51where ρc 1<ρi <ρc 2and −c 0=ρv e (ρi ),i =1,2,(15)such that the linear stability condition (11)or (12)is violated on [ρc 1,ρc 2].Li and Liu [38]showed that the traveling wave solutions of the Payne-Whitham model with non-concave fundamental diagrams are asymptotically stable under small disturbances and under the sub-characteristic condition (11)using a weighted energy method.We are concerned with instability of traffic flow.It will be shown that the instability of traffic occurs at some intermediate density range which contains the unstable region [ρc 1,ρc 2].This is identical to the em-pirical findings for freeway traffic and to the previous numerical simulations.A nonconcave fundamental diagram was used by Kerner and Konh¨a user [26],Jin and Zhang [23]in the study of cluster formation see Fig.1.q (ρ)=5.0461ρ((1+e (ρ−0.25)/0.06)−1−3.72×10−6),(16)Another nonconcave fundamental diagram example isq (ρ)=v f ρ(1−ρ1.4)4,(17)where v f =120km/h,which was used by K¨u hne [30]in his study of the start-and-go waves.The graph of it is similar to the one in Fig.1.Lee et al.[33]adopted a nonconcave fundamental diagram q (ρ)=V 0ρ(1−ρ/ρ0)1+E (ρ/ρ0)θ(18)where V 0=120km/h,ρ0=140veh/km,E =100and θ=4.The graph of it is similar to the fundamental diagram in Fig.1.Moreover,Bando et al.[2]proposed an optimal ve-locity model in which the acceleration of every car is determined by its velocity v n and a desired speed V 0(b n )depending on the headway b n to the car in front d v nd t=a (V 0(b n )−v n)Fig.1.A Nonconcave Fundamental Diagram used by Kerner.T.Li /Physica D 207(2005)41–5145where the optimal velocity V 0(h )=tanh(h −2)+tanh(2)is a monotone increasing function,a is a con-stant called sensitivity which equals the inverse of the reaction time.The evolution of traffic congestion was observed with the development of time.Berg and Woods [4]gave an analogous continuum counterpart of the optimal velocity model which is in good agreement with its discrete version.The stability depends on the choice of parameter a .The fundamental diagram isq (ρ)=ρV (ρ)=ρV 0 1ρ (19)which is nonconcave,x n +1−x n ∈( x min , x max )is unstable,where x min =1and x max =3.The graph of it is similar to the fundamental diagram in Fig.1.3.Derivation of the models Consider the PW model ρt +(ρv )x =0(20)v t +vv x +c 20ρρx =v e (ρ)−vτ.(21)We look for the traveling wave solutions of the PW model (20)and (21),namely,solutions of form (P ,V )(x −ct )=(P ,V )(ξ)where ξ=x −ct is the traveling wave variable.c is the traveling wave speed.We emphasize that we are looking for solutions on a ring road propagating with a negative speed c <0.(22)This reflects the fact that the vehicle clustering travels against the traffic flow.Substituting traveling wave solution (P ,V )(ξ)into (20)and (21),we have −cP +(PV ) =0(23)−cV+VV+c 20P P =v e (P )−Vτ.(24)Integrating (23),we have (V −c )P =Q(25)for some constant Q .Thus Q and c are related by the boundary conditions at ±∞.Under condition (22),Q >0.Solving V in terms of P in (25)and plugging it into (24),we deduce1P 3(−Q 2+c 20P 2)P=v e (P )−c −(Q/P )τ.(26)We denote the coefficient of P in (26)as B (P )=1P3(−Q 2+c 20P 2).(27)On interval (0,1),there is a unique zero of B (P )at P =P 0(Q )=Q c 0provided 0<Q <c 0.Let b (P )=B (P )0(Q ).Then b (P )>0,0≤P ≤1.(28)Denote the numerator of the right hand side of (26)as D (P )=v e (P )−c −QP.(29)Thus the right hand side of (26)is zero if and only if PD (P )=q (P )−(cP +Q )=0,(30)see (4).For clarification of our presentation,we adopt the fundamental diagram q (P )defined in (16)in the rest of the paper,see Fig.1.Lemma 1.For nonconcave fundamental diagram q (P )defined in (16),0<Q <c 0can be chosen such that D (P )=0holds atP =P 0(Q )46T.Li /Physica D 207(2005)41–51Fig.2.q (P )intersects the linear function cP +Q at three points.and at two other points 0<P −(Q )<P +(Q )<1provided c =c (Q )=q (P 0(Q ))−QP 0(Q ).(31)LetI =(min {P −(Q ),P 0(Q )},max {P +(Q ),P 0(Q )}).(32)Then ρi ∈I(33)where 0<ρi <1is the only inflection point of q (ρ),see Fig.2.Proof.First,note that the fundamental diagram q (P )defined in (16)changes concavity at a single point,0<ρi <1.0<Q <c 0can be chosen so that D (P )has three zeros,namely,P 0(Q ),P −(Q ),P +(Q ),see Fig.2.Indeed from (30)we derive (31).Now we prove that the inflection point ρi ∈I ,where interval I is defined in (32).If ρi is not in I ,then q (P )is the convex or concave on interval I .Thus q (P )inter-sects the linear function cP +Q at most at two points on I .From (30),we have that the right hand side of (26)or D (P )has at most two zeros.This contradicts with the fact that D (P )has three zeros.Thus (33)is proved.For simplicity of notation,let us denote P 0(Q ),P −(Q )and P +(Q )as P 0,P −and P +.Let d (P )=D (P )P −P 0.Thend (P )<0,P −<P <P +(34)d (P )>0,0<P <P −,or 1>P >P +(35)T.Li/Physica D207(2005)41–5147 andd(P±)=0.Letf(P)=P+γτd(P)b(P)(36)for someγ>0and for P∈[0,1].It follows from(28)and(34)–(36)thatf(P±)=P±(37) f(P)−P<0,P−<P<P+(38) andf(P)−P>0,0<P<P−,or1>P>P+.(39) f(P)is smooth on interval[0,1].Using the definition of f,Eq.(26)can be written asP =1γ(f(P)−P).(40) (37)shows that P±are two steady states of the nonlin-ear differential Eq.(40).We now propose the following discrete model of trafficflow.DefineP n+1=f(P n),n≥1(41) forδ≤P1≤P+and for0<δ<P−.Then P±are twofixed points of the nonlinear map(41)on interval [δ,P+].Moreover,we have the following lemma. Lemma2.For each0<δ<P−,there is anα0(δ)>0 such that if0<α=γτ≤α0(δ),then f maps the interval [δ,P+]to itself.Proof.First note that whenα=0,f(P)=P which maps the interval[δ,P+]to itself.The conclusion for0<α≤α0(δ)forα0(δ)>0then follows from properties(37)–(39)of f and properchoice ofα0(δ)>0.Continuum model(40)is derived from a nonequilib-rium continuum model under the key physical condi-tions that the fundamental diagram changes concavityand that traffic jams propagate against vehicle trafficflow.The interval I defined in(32)is contained in theunstable region[ρc1,ρc2]defined in(14).In their study of traveling wave solutions to higher order models con-taining viscosity terms,Wilson and Berg[54]proposedthat there is a unique point of inflection of the funda-mental diagram to ensure that chords with negative gra-dient intersect the graph of the fundamental diagram inthree places,see Fig.2.This is the same condition pro-posed in the current paper.Nonconcave fundamentaldiagrams were also used to obtain the cluster solutionsin the unstable region[ρc1,ρc2]defined in(14),Bando et al.[2],Greenberg[14],K¨u hne[30],Kerner and Konh¨a user[26],Jin and Zhang[23],Lee et al.[33].Traffic model(41)is a formal discretization of(40).In other words,we are considering the discretized trav-eling wave solutions.γ>0is the step size of the trav-eling wave variable in the discretization.Remark.Continuous periodic solutions are notobtainable from the profile Eq.(40).This is becauseautonomous oscillations cannot occur in a one dimen-sional system(40).In fact,(38)implies that there areno oscillatory solutions of continuum model(40).Tomodel the complicated dynamic behavior of trafficflowincluding oscillatory behavior,we are led to considerthe discretization(41)of continuum model(40).Underthe same physical setting,a class of such discretemodels can be derived from higher order trafficflowmodels[34,39].The discrete models are similar to(41).4.The dynamicsWe will show that the dynamics of map(41)aregoverned by the logistic map.The dynamics of the lo-gistic map undergoes one stable steady state,a period-2cycle,a period-4cycle and further period-doublingsto cycles of periods8,16,32,...,2n,...,as the bi-furcation parameter increases.The successive bifurca-tions come faster and faster.Ultimately the bifurcation48T.Li/Physica D207(2005)41–51 parameter converges to a limiting value as n→+∞.What happens beyond the limiting parameter value?The answer is complicated:for many values of the pa-rameter,the solution never settles down to anyfixedpoint or a periodic orbit.Instead,the long-time behav-ior is aperiodic and exhibits sensitive dependence oninitial data P1∈[δ,P+]for some0<δ<P−.This isa discrete time version of chaos,see Strogatz[51].Theresults will be compared with the empiricalfindings forfreeway traffic and with the previous numerical simula-tions.Thus the universal period doubling route to chaosgives a justification for observed and simulated trafficinstabilities and some insight into their meanings.Theorem3.Let the initial data P1∈[δ,P+]for some0<δ<P−.Then as0<α≤α0(δ)increases,the dy-namics of nonlinear map(41)undergoes one stablesteady state P−,a period-2cycle,a period-4cycle andfurther period-doublings to cycles of periods8,16,32,...,2n,...,and then chaotic solutions.Proof.Evaluating f atfixed point P=P+and noting(28),(34)and(35),we have thatf (P+)=1+αd (P+)b(P+)>1.(42)Therefore P+is an unstablefixed point of map(41)for all values of0<α≤α0(δ).Similarly,by noting(28),(34)and(35)we havef (P−)=1+αd (P−)b(P−)<1for all values of0<α≤α0(δ).For certain0<δ<P−,there is an0<α1(δ)<α0(δ)such that if0<α<α1(δ),−1<f (P−)<1.(43) Thus P−is a stablefixed point of map(41)and it is the only stable steady state on interval[δ,P+].Therefore the asymptotic behavior of{P n}∞n=1,which was definedin(41)with initial data P1∈[δ,P+],is thatP n→P−as n→+∞.When bifurcation parameterαsatisfiesα1(δ)<α≤α0(δ),thefixed point P−loses its stabilityf (P−)<−1.The resulting bifurcation is called aflip bifurcation. Flip bifurcations are often associated with period-doubling.In the present case,theflip bifurcation does spawn a2-cycle.This is because in a certain bifurca-tion parameter range,there exist two points p and q such thatδ<p<P−<q<P+andf(p)=q,f(q)=p.Thusf(f(p))=p,f(f(q))=qand that p and q are stablefixed points of the sec-ond iteration map f(f(P))on[δ,P+].Therefore a periodic asymptotic state which oscillates between p and q exists.This is a2-cycle.The2-cycle bifur-cates continuously from steady state P−.The2-cycle is stable for bifurcation parameterα<α2(δ)where α1(δ)<α2(δ)<α0(δ).There exists an increasing sequence0<αk(δ)<α0(δ),k=0,1,2,...such that ifαk(δ)<α<αk+1(δ) the asymptotic behavior of{P n}∞n=1exhibits fur-ther period-doublings to cycles of period2k,k= 0,1,2,....As k→+∞,the asymptotic behavior is chaotic.Remark.For the fundamental diagram defined in(16), we found that forδ=0.0113,α1(δ)=15.2540.That is,f maps[0.0113,P+]to itself and that P−is an un-stablefixed point of f ifα>α1(δ).Jin and Zhang[23] numerically solved the PW model(9)and(10)by the Godunov method and found clustering solutions near P−.They also found that the number of clusters,the position,height and the width of each cluster were not predictable.The analysis of large sets of traffic data has revealed the existence of three traffic phases:freeT.Li/Physica D207(2005)41–5149flow,synchronizedflow and wide moving jams, Kerner[24,25],Kerner and Rehborn[27].K¨u hne and co-workers found that there are stable and unstable fixed points and limit cycles of an approximation of the viscous PW model with fundamental diagram (16),the approximation is based on a truncated expansion using the eigenmode expansion from linear stability as a starting point,which suggest that traffic near maximumflow operates on a strange attractor [31].Under certain circumstances,chaotic motion is observed.The unstable traffic patterns are connected with nonlinear stochastics.Free traffic would corre-spond to the point attractor P−.The bifurcation from steady state P−.marks the transition from freeflow to synchronizedflow.Traffic jams are formed when further period-doublings lead to chaotic solutions.The dynamic behavior of map(41)correctly predicts the empiricalfindings for freeway traffic,Daganzo et al.[9],Helbing[19],Kerner[24,25],Kerner and Rehborn [27],Knospe,Santen,Schadschneider and Schrecken-berg[29],Mauch and Cassidy[42],Treiterer and Myers [52],where it was found that speeds,flows and densi-ties exhibit greater variation when measured in moder-ately dense queues as compared with measurements in queues of higher density.Moreover,the oscillations did not affect freelyflowing upstream of a queue’s tail.It was also found that most oscillations formed in moder-ately dense queues while propagating upstream,against theflow of traffic,which agrees with our negative speed condition(22)and that the instability of traffic occurs at some intermediate density range.When the dynamics of map(41)is compared with previous numerical simulations,we see a qualitative agreement.In particular,numerically solving the viscous PW model by the centered Euler scheme, Kerner and Konh¨a user[26]found that if the density of vehicles exceeds some critical value,a small perturba-tion in a homogeneous trafficflow on a circular ring road can grow to a stationary moving cluster.If the density is increased further,the avalanche-like process of cluster formation can start.Therefore,a sequence of clusters,which appear subsequently in space and time, is created.The clusters in this sequence have different amplitudes,different widths,different velocities, and are not situated periodically in space.In their numerical simulations of the viscous PW model(9) and(10),Kerner and Konh¨a user[26]concluded that the chaotic behavior of trafficflow may be linked to the spontaneous appearance of a lot of interacting clusters with different parameters.Different clusters of vehicles,including the cluster of large amplitude moving against theflow,have been observed in the numerical investigations of the cellular automation model by Nagel[43].Microsimulations of road traffic in a ring road setting by Helbing showed that drivers may encounter a traffic jam without obvious cause,phantom jam,or stop-and-go traffic,see[6] (/Traffic/RoadApplet.html) and[19]().Gray and Griffeath [12]analyzed a probabilistic cellular automaton(CA) as a prototype for the emergence of traffic jams at some intermediate density range.Recalling that the bifurcation parameterα=γτ,if we set the size of the discretization of the traveling wave variableγto be a constant,then increasingαis equivalent to increasing1τ.The latter is the sensitiv-ity to the stimulus.Therefore increasing the sensitivity to the stimulus results in richer dynamics of the traffic flow.On the other hand,if wefixτto be a constant,then increasingαis equivalent to increasing the discretiza-tion sizeγ.Therefore increasing the discretization size results in richer dynamics of the trafficflow.This is exactly what the numerical simulations of Bolay[6] showed!The above comparisons show that the numerical simulations agree with the analytical results of dis-crete map(41).We argue that this is because all nu-merical simulations were achieved through solving the discretized differential equations,which is in the same spirit as analyzing the discrete map.5.ConclusionsWe proposed an innovative approach to the nonlin-ear dynamics of trafficflow:a class of discrete models. Discrete model(41)captures the essential features of traffic jams.We were able to identify and formulate the self-organized vehicle clustering and the transition to chaos in traffic.The results were compared with the empiricalfindings for freeway traffic and with the previous numerical simulations.There is a qualitative agreement.Discrete dynamics reflects a new emerging tendency towards utilization of iterative mathematical models to describe the behavior of complex systems.It has50T.Li/Physica D207(2005)41–51became clear from the latest development in discrete modeling that such models have a simpler structure and provide many more possibilities for generating and describing complex non-linear phenomena,including chaotic regimes and fractals,see for example[5]. 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