Exciting traffic jams: Nonlinear phenomena behind traffic jam formation on highways

Transcription

1 PHYSICAL REVIEW E 8, Exciing raffic jams: Nonlinear phenomena behind raffic jam formaion on highways Gábor Orosz,, * R. Eddie Wilson, 2, Róber Szalai, 2, and Gábor Sépán 3, Deparmen of Mechanical Engineering, Universiy of California, Sana Barbara, California 936, USA 2 Deparmen of Engineering Mahemaics, Universiy of Brisol, Brisol BS8 TR, Unied Kingdom 3 Deparmen of Applied Mechanics, Budapes Universiy of Technology and Economics, Budapes H-2, Hungary Received 26 May 29; published 6 Ocober 29 A nonlinear car-following model is sudied wih driver reacion ime delay by using sae-of-he-ar numerical coninuaions echniques. These allow us o unveil he deailed microscopic dynamics as well as o exrac macroscopic properies of raffic flow. Parameer domains are deermined where he uniform flow equilibrium is sable bu sufficienly large exciaions may rigger raffic jams. This behavior becomes more robus as he reacion ime delay is increased. DOI:.3/PhysRevE PACS number s :.4. a, 4.7.Vn, r, 89.4.Bb I. INTRODUCTION Vehicular raffic has been modeled for more han half a cenury bu he precise mechanism for generaion and propagaion of raffic jams is sill no fully undersood. Recenly developed numerical echniques of dynamical sysems can help o shed ligh on he microscopic dynamics underlying he emergen behavior in hese complex sysems. These mehods also allow one o exrac macroscopic flow properies ha can lead o beer undersanding of he fundamenal principles of raffic jam formaion. I is well known ha for sparse raffic here exiss a uniform flow equilibrium where vehicles follow each oher wih he same velociy, while oscillaions may arise when he raffic becomes more dense. One of he ypical oscillaions is a sop-and-go wave; he velociy breaks down and vehicles become densely packed on a secion of he highway and he congesion propagaes upsream as a densiy wave wih a characerisic wave speed of 2 km/h. The congesed regime of finie lengh is enclosed by wo frons ha ravel wih he same velociy: a sop-fron where vehicles ener he congesed regime and a go-fron where hey leave he raffic jam. A driver may encouner such a raffic jam many minues afer i formed and many kilomeers behind he place where i emerged. The name phanom jam is also used for hese congesion waves since drivers canno see any cause of he jam when leaving he congesed regime. There exis many differen models ha are able o reproduce uniform flow as well as sop-and-go waves. However, he ransiion beween hese wo qualiaively differen soluions is sill no clarified. Here, we briefly review he basic deerminisic approaches of raffic modeling and heir proposed mechanisms for jam formaion. Coninuum or macroscopic models ha characerize raffic in erms of densiy and velociy fields use parial differenial equaions PDEs o *gabor@engineering.ucsb.edu re.wilson@brisol.ac.uk r.szalai@brisol.ac.uk Also a Research Group on Dynamics of Vehicles and Machines, Hungarian Academy of Sciences, Budapes H-2, Hungary; sepan@mm.bme.hu describe he ime evoluion of he sysem. The curren saeof-he-ar macroscopic models use hyperbolic PDEs 2,3. In his seup boh he uniform flow and he sop-and-go soluions are marginally sable if hey exis. Consequenly, one may drive he sysem from one soluion o he oher by applying large perurbaions. More precisely, a vehicle needs o come o a hal in order o lead he sysem o he sop-and-go sae: acions have o be as large as he effecs. Car-following or microscopic models describe vehicles as individual eniies. In he simples case he ime evoluion of he sysem is described by ordinary differenial equaions ODEs 4,. These models sugges ha raffic jams form sponaneously when raffic becomes dense enough. More precisely, here exiss a criical raffic densiy, below which he uniform flow is asympoically sable bu i becomes unsable above. Then even iny flucuaions may develop ino sop-and-go waves as hey cascade back along he highway, i.e., iny acions have large effecs. Clearly he hyperbolic PDE and he ODE models sugges wo very differen mechanisms for jam formaion. When driver reacion imes are incorporaed in a carfollowing model, delay differenial equaions DDEs describe he ime evoluion of he sysem 6 8. In his paper, we sudy a simple model wih reacion ime delay and show ha i may reconcile he conflic beween he resuls of he PDE and ODE models by inroducing an exended regime where he uniform flow is asympoically sable bu exciable, so sop-and-go jams may sill emerge. Previously, exciable behavior was deeced only in narrow parameer regimes and papers eiher focused on he mahemaical analysis of he underlying bifurcaions for relaively small numbers of vehicles 9, or conjecured exciabiliy from numerical simulaions,2. Calculaing he bifurcaion srucure explicily for realisic numbers of vehicles and connecing he resuls o macroscopic raffic phenomena are curren ineress in he raffic communiy 3,4. We remark ha one may use nonhyperbolic PDEs o model he dynamics of raffic and also incorporae reacion imes ino coninuum models 6. The effecs of ime delays are also shown o be significan in many oher neworked sysems including neural neworks 7,8 and gene regulaory neworks 9. The mos widely used mehod o invesigae he nonlinear dynamics of raffic is numerical simulaion. Recenly, meh /29/8 4 / The American Physical Sociey

2 OROSZ e al. V. 3 h 7 v i v i+ x i x i+ h i 3 h 7 FIG.. Color online A skech of wo vehicles following each oher is displayed in panel a while he opimal velociy funcion Eq. 6 and is firs derivaive are shown in panels b and c, respecively. ods from dynamical sysems heory normal form calculaions and numerical coninuaion have also been applied o invesigae raffic dynamics 9,2 23. The advanage of hese mehods is ha boh sable and unsable soluions can be sudied. Indeed, unsable moions can hardly be observed in he physical sysem bu hey can influence he emergen behavior by separaing qualiaively differen sable moions. In his paper, we apply numerical coninuaion echniques 24 o reveal he inricae microscopic dynamics underlying jam formaion and o exrac macroscopic flow properies. The layou of he paper is he following: We inroduce he car-following model in Sec. II and review he bifurcaion analysis of he uniform flow in Sec. III. Numerical coninuaion echniques are inroduced in Sec. IV. They are applied o he raffic model in Sec. V where he fundamenal dynamical principles behind exciabiliy are explained. The limi of infiniely many vehicles is discussed in Sec. V A and he spaial moion of waves is sudied in Sec. VB. The full nonlinear dynamics is presened in Sec. VI and we conclude our research and discuss fuure direcions in Sec. VII. II. CAR-FOLLOWING MODELS WITH REACTION TIME DELAY Here, we discuss general modeling issues and inroduce he specific model analyzed in his paper. Assuming idenical vehicles and neares neighbor ineracions, he acceleraion of he i-h vehicle is given by v i = f h i,ḣ i 2,v i 3, where he do sands for differeniaion wih respec o ime, v i is he velociy of he i-h vehicle while h i is he bumpero-bumper disance beween he i-h and he i + -s vehicles also called he headway; see Fig. a. The reacion ime delays, 2, 3 are generally differen, bu someimes, for he sake of simpliciy, hey are considered o be equal o each oher or o be zero. In his paper, we focus on he effecs of. Figure a shows ha he headway can be defined as V. PHYSICAL REVIEW E 8, h i = x i+ x i, where x i is he posiion of he fron bumper of he i-h vehicle and is he vehicles lengh. Taking he ime derivaive one obains he velociy difference ḣ i = v i+ v i, and his kinemaic condiion complees sysem Eq.. One also has o specify boundary condiions. For simpliciy we assume periodic boundary condiions: N vehicles are placed on a circular road of lengh L+N ha yields he algebraic equaion N h i = L, i= where L is called he effecive ring lengh. Using his equaion one may express one headway for example, h N as a funcions of he ohers and so reduce he number of dynamical variables by one. Tha is, sysem Eqs., 3, and 4 can be wrien as a sysem of 2N DDEs. Noice ha he vehicle lengh does no appear in he dynamical equaions. The role of his parameer will be clarified in Sec.V B. Noe ha one may also sudy he sysem on semi-infinie roads where similar paerns can develop as on he ring road for large L and N, bu convecive insabiliies may also need o be handled 2. In previous sudies, he case of arbirarily many vehicles wih weak nonlineariies was sudied by performing normal form calculaions 23 and he case of few vehicles wih srong nonlineariies was invesigaed by numerical coninuaion 2,22. Here, we analyze he case of large number of vehicles wih srong nonlineariies. The resuls are presened for N=33, which is low enough o represen he deailed microscopic dynamics bu high enough o compare he resuls o he case N. Noe ha N is increased such ha L/N is kep consan. We remark ha when reproducing he resuls for larger number of vehicles e.g., N=99, no significan deviaions are found bu he illusraions become more elaborae and so less insrucive. To deermine he funcion f in Eq. one needs o ake ino accoun some general modeling principles; see 3 for deails. In his paper we consider a simple, ye widely acceped model, he so called opimal velociy OV model 4,6,9,2 where f h,ḣ,v = V h v. T In spie of is simpliciy, e.g., i does no depend on he velociy difference ḣ his model is able o reproduce uniform flow as well as sop-and-go waves. The parameer T is called he relaxaion ime and /T is called he sensiiviy. Noe ha T differs from he reacion imes, 2, 3 : he finie relaxaion ime represen he fac ha vehicles have ineria while he reacion imes are explici ime delays in he sysem. In his paper, we consider and 3 = o model he human behavior ha drivers reac o

3 EXCITING TRAFFIC JAMS: NONLINEAR PHENOMENA heir headway wih finie reacion ime and know heir curren velociy insanly. Noe ha 2 does no appear in he equaions since here is no ḣ erm. The funcion V is called he OV funcion ha saisfies he following properies: i i is coninuous, non-negaive, and monoone increasing; ii i approaches he maximum velociy v for large headway, i.e., V h v as h, where he desired speed v corresponds o he speed limi; iii i is zero for small headway, i.e., V h if h,h sop, where h sop is called he sopping disance. We remark ha he resuls presened here are robus agains changes in he OV funcion as long as i saisfies i iii. Before specifying he OV funcion we rescale disances by h sop and rescale ime by h sop /v. Consequenly, velociies including he OV funcion are scaled by v. In his paper we use he rescaled OV funcion =, if h,,,, V h h 3 + h 3, if h 6 ha is shown ogeher wih is firs derivaive in Figs. b and c. Noice ha he rescaled speed limi is. By defining he dimensionless parameers = h sop Tv, = v h sop, he rescaled model can be wrien ino he form v i = V h i v i, ha is closed by he rescaled versions of Eqs. 3 and 4. For approximae realisic values of parameers T,, v, h sop see 2, k PHYSICAL REVIEW E 8, τ = τ = k V h = co k N, ha are parameerized by he frequency R +. Here, k Z + is a discree wave number such ha k N/2. When crossing a sabiliy curve, a Hopf bifurcaion akes place, i.e., a pair of complex conjugae eigenvalues crosses he imaginary axis. The appearing small-ampliude oscillaions are ravelling waves wih frequency and wave number k. In Figs. 2 a and 2 b he sabiliy curves are compared for = and. Noice ha for = he curves become he sraigh lines k k FIG. 2. Linear sabiliy diagrams wihou and wih reacion ime delay. Domains where he uniform flow is linearly sable are shaded. Panels a,b show sabiliy chars on he (V h, )-plane ha are ransformed o he h, -plane in panels c,d. The arrows represen he increase of wave number k. k V (d) h III. BIFURCATIONS OF THE UNIFORM FLOW 2 = 2 cos k V h N cr, The sysem Eqs. 3, 4, and 8 possesses a uniform flow equilibrium h i h = L/N, v i v = V h, 9 for i=,...,n ha may lose sabiliy when he parameer h is varied. To describe wha kind of paerns can appear in sysem Eqs. 3, 4, and 8 we briefly summarize he resuls of he linear sabiliy analysis shown in Linearizing he sysem abou he uniform flow equilibrium Eq. 9, using rial soluions proporional o e, C and considering he criical eigenvalues h cr = i, one can obain he sabiliy curves V h cr = 2 cos sin k k, N N and as. For he curves approach verical asympoes locaed a k V h cr = k N 2 sin k, N 2 when N. Tha is for large enough he curves are conained by he regime V 2, 4. The sabiliy curves are ordered such ha he mode wih lowes wave number k= i.e., wih he longes wavelengh gives he sabiliy boundary and he curves wih k lead o furher insabiliies as k is increased. This means ha he uniform flow loses is sabiliy o ravelling waves wih long wavelengh firs and hen waves wih shorer and shorer wavelengh may show up, oo. This behavior is illusraed in Figs. 2 a and 2 b where he uniform flow is linearly sable 462-3

4 OROSZ e al. in he shaded domains and he arrows show he increase of wave number k. The ordering of sabiliy curves is preserved for any feasible model Eq. wih zero delays = 2 = 3 =; see 3. In he case of he delayed model Eq. 8 we found ha he order is also kep for. Using he derivaive of he OV funcion Eq. 6 in Fig. c, one may ransform he sabiliy chars from he V h, -plane Figs. 2 a and 2 b o he h, -plane Figs. 2 c and 2 d. For = here is a single curve for each wave number k in he h, -plane such ha he wave number increases from ouside o inside. For sufficienly large values of he uniform flow is sable for any value of he average densiy h. For = here are wo curves for each wave number k wih verical asympoes, i.e., for any values of he uniform flow is sable for large h sparse raffic and small h dense raffic and i is unsable for inermediae values of h. One may check ha his behavior persiss as N is increased and he N=33 case is a good represenaion of he large N case. Noe ha in he sable regime for small h vehicles ravel wih he low velociy or sand on he freeway when h. We remark ha for 3 2 7/3 he sabiliy char is similar o he = case bu he maxima of he curves are larger. A =3 2 7/3.9 he curve for wave number k= becomes unbounded above. When increasing furher, curves for higher and higher wave numbers become unbounded above finishing wih he curve for k = N/ 2 a =3 2 /3.93. This can be seen by calculaing he maximum of V h from Eq. 6 and using Eq. 2. In he lieraure, sabiliy diagrams are ofen ploed using he dimensional parameer /T insead of. Then in he nondelayed case he maxima of he closed sabiliy curves are proporional o v bu o obain sabiliy curves ha are unbounded above he limi v needs o be aken. In he delayed case, he curves are unbounded above even for moderae values of v. When h is close o h cr, he small-ampliude ravelling wave soluion can be wrien ino he form v i = v + 2 v amp cos 2 k N i +, 3 for i=,...,n. From Eq., one can deermine a dispersion relaion as he funcion of k and calculae he spaial wave speed c = v h V h cr O k 2. 4 N A leading order he delay only influences he wave speed by changing he quaniy V h cr. To deermine he peak-o-peak ampliude of oscillaions v amp in Eq. 3 nonlineariies has o be considered. Using hird-order approximaion of nonlineariies and carrying ou Hopf normal form calculaions, one may obain he ampliude PHYSICAL REVIEW E 8, v amp =2 Re h cr h h cr, where h cr describes he speed of crossing he imaginary axis by he criical eigenvalue h cr =i as he parameer h is varied. For he Hopf bifurcaions are supercriical small ampliude oscillaions appear afer he pair of complex conjugae eigenvalues crosses he imaginary axis, while for hey are subcriical oscillaions appear before he eigenvalues cross he imaginary axis. For k= supercriicaliy resuls in sable small-ampliude waves while subcriicaliy gives unsable ones. For k boh cases lead o unsable waves as will be discussed in Secs. V and VI. I was shown in 23 ha when he delay is included in he model, he bifurcaions are robusly subcriical and was given in a closed form for arbirary N. Indeed, he wave speed Eq. 4 and he ampliude Eq. are only good approximaions for small-ampliude oscillaions when h is close o h cr. To describe he behavior far from he bifurcaion poins one needs o use numerical mehods. IV. NUMERICAL CONTINUATION TECHNIQUES In his secion, we provide he reader wih some deails abou he numerical coninuaion echniques ha are he principal ools of invesigaion in he subsequen secions. Those who are familiar wih such echniques may skip his secion. The main advanage of hese mehods, compared o numerical simulaion, is ha boh sable and unsable saes can be sudied. For ODEs one can use he package AUTO 26 while for DDEs he packages DDE-BIFTOOL 27 and PDDE-CONT 28 are available. We describe he capabiliies of hese packages ha are exploied in his paper; for more deails see 24. Numerical coninuaion packages are able o follow branches of equilibria and periodic soluions as a parameer is varied. Sabiliy informaion is compued along he branches and bifurcaion poins where he sabiliy of soluions changes are deeced auomaically. For example, considering he DDE Eqs. 3, 4, and 8 one may fix and, vary he parameer h and sudy he uniform flow equilibrium and he oscillaions arising a he Hopf bifurcaion poins. Subsiuing he uniform flow equilibrium ino he DDE resuls in algebraic equaions. For a chosen parameer h, hese can be solved numerically by using an iniial guess and he Newon-Raphson mehod. Then he resul can be used as an iniial approximaion when solving he same se of algebraic equaions for he slighly changed parameer h +dh. By coninuing his process, a branch of equilibria is obained as a funcion of he bifurcaion parameer. Indeed, his is a numerical represenaion of Eq. 9. To deermine he sabiliy of equilibria he linearizaion of he DDE is considered and he corresponding complex eigenvalues C are compued numerically. There are infiniely many eigenvalues bu only finiely many are locaed on he righ side of a chosen verical boundary in he complex plane. The uniform flow equilibrium is sable when all 462-4

5 EXCITING TRAFFIC JAMS: NONLINEAR PHENOMENA eigenvalues are locaed on he lef-half complex plane. Hopf bifurcaions of equilibria are deeced when a pair of complex conjugae eigenvalues crosses he imaginary axis a i. To be able o follow he branches of arising periodic soluions he oscillaions are represened on a finie mesh wih a small number of so-called collocaion poins in beween he mesh poins. On each mesh inerval he soluion is given by a polynomial and he degree of he polynomial is equal o he number of collocaion poins. In he close viciniy of he Hopf bifurcaion poins he firs harmonics wih frequency cf. Eq. 3 can be used as iniial approximaion for he periodic orbis and hese can be correced by he Newon- Raphson mehod. Then, similarly o equilibria, branches of periodic orbis can be coninued by using he soluion a a cerain branch poin as an iniial approximaion for he nex branch poin. To deermine he sabiliy of oscillaory soluions he soluion operaor of he DDE is discreized and he eigenvalues of he resuling large marix, he Floque mulipliers, are calculaed. There are infiniely many Floque mulipliers bu only finiely many are locaed ouside a chosen neighborhood of he origin. The periodic soluions are sable when all Floque mulipliers are locaed inside he uni circle excep a rivial muliplier a + corresponding o coninuous ranslaional symmery along he periodic orbi. Fold bifurcaions of periodic soluions are deeced when an addiional real Floque muliplier crosses he uni circle a +. Since he branch folds back in his case, arclengh paramerizaion of he curve is implemened. I is also possible o fix only he parameer and vary he remaining wo parameers h and while prescribing he addiional consrain ha a bifurcaion occurs. For example, considering he uniform flow equilibrium and assuming ha Hopf bifurcaion occurs i.e., here exiss a pair of purely imaginary eigenvalues i, he Hopf bifurcaion curves in he h, -plane can be raced numerically. More imporanly, considering periodic soluion and assuming ha fold bifurcaion occurs i.e., here exis a double Floque muliplier a +, fold bifurcaion curves can be raced in he h, parameer plane. We remark ha for DDEs his can only be achieved by using he recenly developed package PDDE-CONT 28. We emphasize ha he applicaion of coninuaion packages is a much more efficien way of exploring parameer space han performing mass ensemble simulaion of he iniial value problem. This is especially rue for DDEs where he iniial condiions are funcions in he inerval,. V. NONLINEAR DYNAMICS BEHIND EXCITABILITY In Sec. III, we deermined ha he sabiliy curve for k= gives he sabiliy boundary. In his secion we focus on he nonlinear behavior arising from his long wavelengh insabiliy by using he echniques described in Sec. IV. These mehods allow us o reveal he inricae microscopic dynamics of he sysem including hidden unsable oscillaions. Moreover, macroscopic properies of he flow can also be exraced h.3.2. v amp q. PHYSICAL REVIEW E 8, τ = τ = Firs we fix and and vary h, i.e., we sudy he sysem along he horizonal line = in Figs. 2 c and 2 d. In Figs. 3 a and 3 b he peak-o-peak velociy ampliude v amp = max /h D C B.8 v amp h v i min v i, 6 of he appearing oscillaions are shown as a funcion of he average headway h. Noe ha v amp is he same for every i since he moion of vehicles is idenical excep a shif in ime. We will exploi his propery when defining quaniies like he flux furher below. Solid green ligh grey and dashed red dark grey curves correspond o sable and unsable saes, respecively. The hick curve along he horizonal axis represens he uniform flow ha is sable for small and large values of h and unsable for inermediae values. For he oscillaory soluions v amp, as shown by he hin curves. The Hopf bifurcaion poins where he uniform flow loses and gains sabiliy are shown as black sars. The Hopf bifurcaions are subcriical, ha is, he appearing small ampliude oscillaions are unsable. This corresponds o in Eq. ha approximaes he peak-o-peak ampliude close o he Hopf bifurcaion poin. The branches of oscillaory soluions fold back for large ampliude resuling in sable oscillaions. These fold bifurcaion poins are marked by blue s q E A B C A D E. (d) /h FIG. 3. Color online Bifurcaion diagrams for he lowes wave number k= wihou and wih reacion ime delay in case of =. In panels a,b he peak-o-peak velociy ampliude v amp is ploed as a funcion of he average headway h while in panels c,d he flux q is ploed as a funcions of he average densiy /h. The righ side of panel a corresponds o he lef side of panel c and he same holds for panels b and d. Thick curves correspond o he uniform flow and hin curves correspond o oscillaory ravelling wave soluions. Sable and unsable saes are shown as solid green ligh grey and dashed red dark grey curves, respecively. The black doed curve in panels c,d corresponds o he approximaion of he flux for N. Hopf and fold bifurcaions are denoed by black sars and blue s, respecively. The black dos A-E in panels b,d correspond o he ime profiles in Fig

6 OROSZ e al. v h The bifurcaion diagrams show ha here exis hree qualiaively differen behaviors: i In he regimes o he lef of he lef fold poin and o he righ of he righ fold poin he only linearly sable sae is he uniform flow soluion and consequenly his sae is globally sable. ii In he regime beween he wo Hopf poins he uniform flow is linearly unsable and he only linearly sable sae is he large-ampliude oscillaory soluion ha is consequenly globally sable. iii In he regimes beween Hopf and fold poins boh he uniform flow and he large-ampliude oscillaory soluions are linearly sable and hey are separaed by an unsable oscillaory soluion. This means ha he sysem is bisable: depending on he iniial condiions eiher he uniform flow or he oscillaory soluion is approached. Observe ha he bisable regimes become much more pronounced in he delayed case; he delay makes his behavior very robus. In fac, for cerain OV funcions ha differ from Eq. 6 he 2. v h T jam 2. v h T jam 2. v h T jam (d) v h (e) 2. v. 2. h (h,v ) v. 2. h (h,v ) v. (h,v ) (h +,v + ) (f) (h,v ) (h +,v + ) (g) (h,v ) (h +,v + ) (h) 2. h (h,v ) FIG. 4. Oscillaions for wave number k= corresponding o he black dos A-E in Figs. 3 b and 3 d. In panels a e he velociy of he firs vehicle is shown as a solid curve scaled on he lef while he headway of he firs vehicle is shown as a dashed-doed curve scaled on he righ. Panels f h show he oscillaions in sae space. Noice ha he small-ampliude unsable oscillaions a,e are similar o homoclinic orbis, while he large-ampliude sable oscillaions b d are similar o heeroclinic orbis. PHYSICAL REVIEW E 8, bisabiliy may disappear for = bu i always exiss for large enough ; see 23. To reveal he deails of he nonlinear dynamics we marked he poins A-E along he oscillaory branch in Fig. 3 b. Poins A, B are in he bisable regime on he righ h =2.9, poin C is in he regime where he uniform flow is linearly unsable h =2. and poins D,E are in he bisable regime on he lef h =.. Figures 4 a 4 e shows he corresponding ime profiles for velociy solid curve and headway dashed-doed curve for he firs vehicle, while panels f h depic he periodic orbis in sae space. These soluions preserve he ravelling wave feaures: he ime profiles of he oher vehicles can be obained by shifing he oscillaions wih T p /N where T p is he period of oscillaions. The small-ampliude unsable oscillaions in panel a consiss of a plaeau of consan velociy ha is inerruped by a dich where he velociy is reduced for a shor ime he driver aps he brake shorly. The velociy along he plaeau is close o bu slighly higher han he velociy of he uniform flow Eq. 9. This can be observed in panel f where he corner of he small-ampliude periodic orbi corresponds o he plaeau while he do a h,v represens he uniform flow. The corner and he do are very close and he do is locaed inside he limi cycle. The unsable periodic orbi is similar o a homoclinic orbi and for N his becomes a homoclinic orbi since he lengh of he plaeau goes o infiniy. Moving poin A along he unsable oscillaory branch in Fig. 3 b from lef o righ he velociy is reduced more and more during he dich reaching zero a he fold poin. The small-ampliude unsable oscillaions in panel e are similar o hose in panel a bu here he velociy plaeau is inerruped by a hump where he velociy is increased for a shor period of ime. Again, he velociy along he plaeau is close o he values of he uniform flow Eq. 9 and he unsable periodic orbi is close o a homoclinic orbi as displayed in panel h. Moving poin E along he unsable oscillaory branch in Fig. 3 b from righ o lef he velociy increases more and more during he hump. The large-ampliude sable oscillaions shown in panels b d consis of a high-velociy plaeau and a low-velociy plaeau ha are conneced by a sop-fron where vehicles decelerae and a go-fron where cars accelerae. The corresponding ravelling wave is a sop-and-go wave. The oscillaions are shown in sae space in panels f h where he corners of he large-ampliude periodic orbis correspond o he velociy plaeaux. The sable periodic orbi is similar o a heeroclinic orbi and for N i becomes a heeroclinic orbi since he lengh of he plaeaux become infinie. Observe ha he period of oscillaions does no change significanly beween panels a e. Moving along he sable oscillaory branch in Fig. 3 b from righ o lef he velociy along he plaeaux and he shape of he frons do no change significanly bu he fracion of ime spen in he low-velociy sae increases. To quanify his change we inroduce T jam, he ime inerval corresponding o v i /3. Thus one can define he flux 462-6

7 EXCITING TRAFFIC JAMS: NONLINEAR PHENOMENA q = min v i T jam + max v i min h i T p max h i T jam, 7 T p where T p is he period of oscillaions. This quaniy gives only an approximae value of he flux bu i is a good esimae for large N when he widh of he plaeaux is much smaller han he widh of he frons. Indeed, he choice /3 is arbirary bu choosing any oher velociy beween and only resuls in small quaniaive differences. In Figs. 3 c and 3 d he so-called fundamenal diagrams show he flux q as a funcion of average densiy =/h. Noe ha he righ side of panel a corresponds o he lef side of panel c and vice verse, and he same holds for panels b and d. For he uniform flow one may calculae he flux as q = v h = V h h = V, 8 ha gives he hick concave curve. Considering he spaial wave speed Eq. 4 for N k and h =h cr one obains c V h cr h cr V h cr = dq d =/h cr, 9 which is he derivaive of he concave curve a /h cr where he flow loses and gains sabiliy. This is in agreemen wih he kineic heory of linear waves 29. From he fundamenal diagram one may noice ha c changes sign as one increases he delay, i.e., he unsable small-ampliude waves propagae backward for = and propagae forward for =. However, his does no persis for large-ampliude waves ha always propagae upsream as will be demonsraed in Sec. VB. Bisabiliy can be observed in he fundamenal diagrams and his behavior becomes very robus for he delayed case. Observe ha he flux for unsable oscillaions is almos he same as he flux of he uniform flow since he velociy changes in a very shor ime inerval in Figs. 3 a and 3 e. This vindicaes ha sudying only he fundamenal diagram is no adequae o reveal he mechanism behind jam formaion in he bisable regime bu one needs o pay close aenion o driver behavior. We remark ha o race he sable par of he fundamenal diagram one may use numerical simulaions and differen vehicle couning mehods 3. A. Large N Limi Now we compare our resuls o he large N limi ha is aken as N while fixing h =L/N. In paricular, we are ineresed in where he fold poins are locaed for large N. In 22 i was shown ha for inermediae values of he headway he sysem reaches he large N limi fas, ha is, no quaniaive differences can be found for N 2. In his regime we may inroduce h = min v = min and realize ha h i, v i, h + = max v + = max h i, v i, 2 h h + PHYSICAL REVIEW E 8, τ = τ = v V h, v + V h +. 2 These quaniies do no change significanly along he sable oscillaory branch excep when he parameers are close o he fold bifurcaion poins. Furhermore, calculaing T jam along he branch we obain ha T jam T p h+ h h + h. Subsiuing Eqs ino Eq. 7 gives h q V h h + h h h + h + V h+ h h h + h + h, ha corresponds o he kineic heory of nonlinear waves 29. This allows us o calculae he flux along he sable oscillaory branch o any average headway h or average densiy /h using he quaniies h and h + ha are deermined from a single inermediae value of h. Indeed, he larger N is he more accurae are he approximaions Eqs In Figs. 3 c and 3 d he flux Eq. 23 is ploed as a doed black curve ha esimaes he solid green ligh grey branch of oscillaions wih high accuracy excep close o he fold poins. The congesed sae h,v and he freeflow sae h +,v + are shown ogeher wih he uniform flow equilibrium h,v in Figs. 4 f 4 h. InheN limi hese quasiequilibria are conneced by heeroclinic orbis. Formula 22 also reveals wha happens a he fold bifurcaion poins. A he righ fold poin he lengh of he low velociy plaeau goes o zero T jam : almos all vehicles ravel wih velociy v + and hey are separaed by h +. Tha is, he fold bifurcaion occurs when h h +. Similarly, a he lef fold poin he lengh of he high velociy plaeau goes o zero T jam T p : almos all vehicles ravel wih velociy v and separaed by h and so h h. Again, hese approximaions become more accurae as N is increased. h h + FIG.. Color online Phase diagrams for wave number k= comparing he nondelayed and delayed cases. The uniform flow is linearly sable in he ligh and dark grey domains and unsable in he whie domain ha is bounded by he solid black Hopf bifurcaion curves. Large ampliude oscillaions exis in he whie and dark grey regimes beween he solid blue fold bifurcaion curves. Consequenly, he sysem is bisable in he dark grey domains. Dashed blue curves indicae he boundaries of bisabiliy in he N limi and collisions occur below he horizonal dashed-doed lines. Noice ha he bisable regimes are much more exended in he case wih delay. h 462-7

8 OROSZ e al. To check how close he N=33 case is o he N case, we locae he fold bifurcaions and also he limis h and h + for differen values of. The resuls are shown in Fig.. The solid black curves separaing he whie and dark grey regions are he Hopf bifurcaion curves for k=; cf. he ouermos curves in Figs. 2 c and 2 d. The solid blue curves separaing he ligh and dark grey regions are he fold bifurcaion curves and hey are deermined by wo-parameer numerical coninuaion. The dashed blue curves show h and h +. These are calculaed using Eq. 2 from one-parameer numerical coninuaion when is varied and h is fixed. The fold curves are very close o he h and h + curves, which vindicaes ha he large N limi is well approximaed by he N=33 case. In he large N limi, one may also deermine he boundary of collisions from he h curve. Collisions occur for negaive h, ha is, for values ha are below he poin where he h curve inersecs he verical axis. This boundary is shown Fig. as dashed-doed horizonal line. Noice ha he collision regime is more exended in he delayed case. Regimes of globally sable uniform flow ouside he fold curves, globally sable sop-and-go oscillaions beween he Hopf curves and bisabiliy enclosed by Hopf and fold curves are shaded as ligh grey, whie and dark grey, respecively. Observe ha he bisable regime is much larger in he delayed case. For = i shrinks o zero and disappears as increases while for i persiss even for large, i.e., he delay makes he bisabiliy robus. We emphasize ha in he bisable regime he sysem is exciable: small perurbaions decay while large perurbaions lead o sop-and-go waves. B. Bisable Wave Dynamics In his secion we use numerical simulaion o invesigae he spaioemporal dynamics induced by bisabiliy. We focus on he regime on he righ of Figs. 3 b and b and sudy he exciable dynamics in deail. We demonsrae ha one may obain qualiaively differen emergen behaviors by changing only he iniial condiions. We also quanify he criical iniial condiions ha separae differen behaviors. Recall ha he iniial condiions are funcions on he inerval, ha are chosen o be consan funcions here. I was shown analyically in 23 ha imposing large enough sinusoidal spaial inhomogeneiy on he sysem may lead o sop-and-go waves. However, he ime profiles in Fig. 4 a sugges ha even localized perurbaions can rigger raffic jams. To es his idea we se he iniial condiions as follows: all vehicles are a he uniform flow equilibrium Eq. 9 excep one which has is velociy reduced by v per and is headway increased by h per. This seup mimics he effec ha a driver aps he brake for a shor ime inerval. More precisely, we assume ha he seleced driver has deceleraed wih a br for he ime inerval T br, and so is velociy is reduced by v per = a br T br, 24 and is headway is increased by h per = 2 a br T 2 br. 2 We fix he braking inerval T br and vary he braking srengh a br o deermine he criical perurbaion needed o rigger B v amp( ) v per(, ) A h 4 PHYSICAL REVIEW E 8, sop-and-go raffic jams. In Fig. 6 a he peak-o-peak velociy ampliude Eq. 6 is compared wih he criical velociy perurbaion v per ; cf. he righ side of he bifurcaion diagram in Fig. 3 b. Similarly, in Fig. 6 b he peak-o-peak headway ampliude h amp = max h 4 h i min h i, 26 is compared wih he criical headway perurbaion h per. Circles and diamonds correspond o T br =. and T br =7., respecively. The resuls vindicae ha sufficienly large localized exciaions can rigger sop-and-go jams and ha he criical exciaion increases wih he ampliude of unsable oscillaions. This demonsraes ha a single driver may drive he whole sysem o he sop-and-go sae even hough he uniform flow is sable agains small perurbaions. Indeed, here is no perfec mach beween he criical exciaion and he ampliude since he localized perurbaions do no exacly place he sysem o he unsable manifold of he limi cycle bu simply o one or he oher side of he sable manifold of he limi cycle. Sill, he qualiaive dynamics of he sysem is deermined by he unsable oscillaions. To illusrae he spaioemporal dynamics we fix h =2.9, which corresponds o he dos A,B in Figs. 3 b and 6 and o he oscillaion profiles in Figs. 4 a, 4 b, and 4 f. The resuls are shown in Fig. 7 where he velociies of every hird vehicle are depiced in he op panels a,b and he corresponding posiions are displayed in he boom panels c,d. The iniial condiions are chosen according o Eqs. 24 and 2, i.e., we se T br =., a br =.6 v per =.3, h per =.76 on he lef panels a,c and T br =., a br =.6 v per =.3, h per =.762 on he righ panels b,d. The laer case corresponds o he circle in Fig. 6 a h =2.9. The small differences in iniial condiions lead o very differen emergen behavior. On he lef he perurbaions decay and he sysem reurns o he uniform flow equilibrium, while on he righ perurbaions are amplified as hey cascade back along he road and a sop-and-go wave develops. Now we focus on he developmen of he sop-and-go wave in Figs. 7 b and 7 d. For he small ampliude wave propagaes downsream corresponding o c in Eqs. 4 and 9 bu as he ampliude increases a backward propagaing sop-and-go wave emerges. We deec when he B h amp( ) h per(, ) FIG. 6. Color online Comparing he peak-o-peak ampliudes Eqs. 6 and 26 of unsable oscillaions wih he criical srengh of localized perurbaions Eqs. 24 and 2 for he velociy a and he headway b in case of k=. The circles and diamonds correspond o braking inervals T br =. and T br =7.. A 462-8

9 EXCITING TRAFFIC JAMS: NONLINEAR PHENOMENA PHYSICAL REVIEW E 8, v i. v i v amp τ = τ = G.8 v amp.6.4 F x i x i 2 4 (d) h h (d) FIG. 7. Color online Demonsraion of bisabiliy in spaceime. In panels a,b he velociies of every hird vehicle are shown while panels c,d display he corresponding posiions. The rajecory of he firs vehicle is emphasized as black. Observe ha he small difference in iniial condiions beween a,c and b,d leads o large differences in he emergen sae. In panel d he sop-fron lower curve and he go-fron upper curve are highlighed as red and green, respecively; see Fig. 8 a for zoom-in. Noice he differen spaial wave speed of he developing and he fully developed sop-and-go wave. velociy of a vehicle drops below /3 and his gives he sopfron in space lower curve, highlighed as red. We also deec when he velociy of a vehicle exceeds /3 and his gives he go-fron in space upper curve, highlighed as green. These frons separae he congesed regime where he velociy is below /3 and he free-flow regime in space. Deecing hese velociy crossings provides us wih discree poins in space ime bu hese are lined up o visualize he fron moion. The ime evoluion of he sop-and-go wave can be described hrough he moion of he corresponding frons. As he wave develops here exis wo differen sopand-go regimes wih differen fron behavior and wave 3 x i x i h FIG. 8. Color online Zoom of Fig. 8 d is shown in panel a when all rajecories are displayed. Panel b shows he same siuaion bu wih nonzero vehicle lengh. The boom and he op of a fa rajecory correspond o he moion of he fron and rear bumper of a vehicle. The sop-fron lower curve and he go-fron upper curve are highlighed as red and green, respecively. Observe ha he vehicle lengh alers he wave propagaion speed significanly. h h FIG. 9. Color online Bifurcaion diagrams for = a,b and phase diagrams c,d comparing he cases wihou and wih reacion ime delay for all wave numbers. The same noaion is used as in Figs. 3 and. In he regimes beween he ouermos Hopf curve he k-h fold curve couning from ouside o inside one may excie k sop-and-go waves. The black dos F,G in panel b correspond o he ime profiles in Fig.. speed. For 2 he sop-fron and he go-fron propagae upsream wih differen velociies such ha hey move away from each oher, i.e., he congesed region exends. For 2 he frons propagae wih he same velociy and he sysem reaches he sae corresponding o Figs. 4 b and 4 f. The regime where he velociy is below /3 in Fig. 4 b corresponds o he congesed regime in Fig. 7 d. In fac, a a given momen in ime mos vehicles are eiher a he congesed sae h,v or a he free-flow sae h +,v + and only a few cars ravel wih velociy beween v and v + as shown by he zoom in Fig. 8 a, where he rajecories of all vehicles are displayed. In he regime 2, he fron velociies are well approximaed by c = h+ v h v + h + h, 27 ha is obained from kineic heory of nonlinear waves 29. For he parameers considered here we have c=.67 ha fis very well o Fig. 7 d. So far we only considered vehicles of zero lengh, i.e., = in Eq. 2. For nonzero he qualiaive dynamics do no change since he dynamical sysem Eqs. 3, 4, and 8 does no conain his parameer. However, he parameer alers he spaial wave propagaion speed of he sop-gowave significanly according o c = c + h. 28 This is demonsraed in Fig. 8 b where he same spaioemporal plo is shown as in Fig. 8 a bu for =.3. This h 462-9

10 OROSZ e al. v h indicaes ha considering lorries insead of auomobiles resuls in significanly faser wave propagaion. VI. TRIGGERING MULTIPLE JAM FORMATIONS In his secion, we invesigae oscillaions corresponding o higher wave numbers k. I was shown in Fig. 2 ha such oscillaions arise when crossing Hopf bifurcaion curves in he linearly unsable parameer domain. The resuling small ampliude oscillaions are always unsable bu furher bifurcaions may occur as he ampliude increases. The velociy ampliude of oscillaions are shown in Figs. 9 a and 9 b for = where he same noaion is used as in Figs. 3 a and 3 b, excep ha unsable branches are shown as solid red dark grey curves. The ouermos branch belongs o k= and k increases from ouside o inside. We found ha he oscillaory soluions are always unsable for k. When he Hopf bifurcaion is subcriical he oscillaory branch undergoes a fold bifurcaion similarly o he k= case bu i does no gain sabiliy. In he non-delayed case only he firs 7 branches appear. The Hopf bifurcaions are subcriical for he ouer branches k=,...,3 and supercriical for he inner branches k=4,...,7. In he delayed case all 6 branches are presen and he Hopf bifurcaions are subcriical for all branches. This robus subcriicaliy leads o exended regions where he sable uniform flow coexis wih oscillaions belonging o differen wave numbers. Figures 9 c and 9 d depic he Hopf and fold curves in he h, -plane where he same noaion is used as in Figs. a and b. These figures demonsrae ha, similarly o he bisable regions, he regions of coexisence are much more pronounced in he delayed case. The ouer fold curves are almos equally spaced: hey divide he bisable domain ino regions of equal widh and he complexiy of he dynamics he number of unsable oscillaions increases when moving from ouside o inside. To undersand he role of his complexiy in he emergen behavior we sudy he oscillaions for k in deail. We choose he k=4 case o demonsrae he dynamics bu he resul are qualiaively he same for any k wih small k/n. We marked he poins G,F along he k=4 branch in Fig. 9 b a h =2.9 and plo he corresponding oscillaions in 2. v h v. (h,v ) (h +,v + ) 2. h (h,v ) FIG.. Oscillaions for wave number k=4 corresponding o he black dos F,G in Fig. 9 b. Noaion and ime scales are as in Figs. 4 a, 4 b, and 4 f G v amp( ) v per(, ) F h 4 PHYSICAL REVIEW E 8, h 4 Fig.. The same noaion and ime scales are used as in Figs. 4 a, 4 b, and 4 f, ha is, he period of oscillaions for k is abou one k-h of he oscillaions for k=. In general he period is proporional o N/k. Furhermore, here are remarkable similariies beween he k= and k cases considering he shape of he periodic orbis. The velociy of he small-ampliude homoclinic-like oscillaions plaeaus a approximaely he same value and only differ in how much he velociy is reduced during he diches. For he large ampliude heeroclinic-like oscillaions he plaeaux and he frons look almos idenical. Noice ha adding up he ime inervals when he velociy is below /3 for k approximaely gives T jam for k=, ha is, he ime spen in congesion is preserved for small k. One may sudy he sabiliy of hese soluions in deail and find ha while he small ampliude oscillaions are srongly unsable here exis a Floque muliplier ha is much larger han in magniude, he large ampliude oscillaions are only weakly unsable he larges Floque muliplier is ouside he uni circle bu very close o i. Due o his weak insabiliy he sysem can say in he neighborhood of large ampliude oscillaions for long bu sill finie ime. This sugges ha applying sufficienly large localized perurbaions a k sies one may excie large ampliude oscillaions of wave number k. To observe such exciable behavior he parameers need o be chosen from he regimes beween he ouermos Hopf curve and he k-h fold curves couning from ouside o inside in Fig. 9. Similarly o he k= case, he criical value of localized perurbaions can be deermined by numerical simulaions for k. For example, considering k=4 we iniialize he sysem ino he uniform flow bu reduce he velociy and increase he headway of four differen vehicles by v per and h per according o Eqs. 24 and 2. Figure compares he peak-o-peak ampliude of oscillaions for k=4 cf. he righ side of Fig. 9 b wih he criical perurbaions needed o rigger four sop-and-go jams. The resuls are presened for T br =. circles and T br =7. diamonds. Observe ha he exciable behavior is qualiaively similar o he k = case; cf. Fig. 6. To illusrae his exciabiliy in space-ime, we fix h =2.9 ha corresponds o he dos F,G in Figs. 9 b and and o he oscillaion profiles in Fig.. Using Eqs. 24 and 2 we se he iniial condiions as T br =., a br =.8 v per =.4, h per =. corre G h amp( ) h per(, ) FIG.. Color online Comparing he peak-o-peak ampliudes Eqs. 6 and 26 of unsable oscillaions wih he criical srengh of localized perurbaions Eqs. 24 and 2 for he velociy a and he headway b in case of wave number k=4. The circles and diamonds correspond o braking inervals T br =. and T br =7.. F 462-

11 EXCITING TRAFFIC JAMS: NONLINEAR PHENOMENA 9 x i sponding o he circle in Fig. a h =2.9. The spaioemporal plo in Fig. 2 a shows he posiion of every hird vehicle and he deeced sop-frons lower red curves and go-frons upper green curves. These frons enclose he congesed regions where he velociy is below /3. Noe ha he frons are deeced simulaneously as ime progresses and i requires furher daa processing o separae he poins for individual frons. The frons ravel wih almos he same velociy which is well approximaed by c in Eq. 27. To reveal he slow dynamics we eliminae his consan velociy moion in Fig. 2 b and show ime on a logarihmic scale. The relaive velociy of he frons wih respec o each oher is usually very small bu here are abrup changes when a sop-fron and a go-fron of a raffic jam approaches each oher and he congesed regime disperses. Through hese dispersions he number of raffic jams gradually decreases o one, ha is, he sable sop-and-go wave for k= is approached. Noice ha he sum of he widh of congesed regimes is approximaely consan for any k, i.e., he number of vehicles in congesion is preserved for small k. We remark ha one may rigger k raffic jams using localized perurbaions ha are no evenly disribued. Sill, he slow dynamics of frons remain qualiaively similar o Fig. 2 b, excep ha merging of raffic jams may occur as well when he sop-fron of a raffic jam approaches he go-fron of he neighboring jam. I may be an ineresing fuure research o model he spaioemporal fron dynamics explicily. VII. CONCLUSION AND DISCUSSION 9 x i -c FIG. 2. Color online Demonsraing he dynamics of four riggered raffic jams in space-ime. In panel a he same noaion is used as in Fig. 7. In panel b he slow dynamics of he arising raffic jams are shown when he consan speed moion is eliminaed. A jam disperses when is sop-fron lower red curve and go-fron upper green curve mee wih each oher. Observe ha he sum of widh of he congesed regimes is approximaely consan. 6 3 PHYSICAL REVIEW E 8, In his paper we sudied a car-following model wih driver reacion ime delay. We found ha he dynamics of he relaed DDEs are robusly exciable. Exended regimes are idenified in parameer space where he uniform flow is linearly sable bu raffic jams can be riggered by large enough localized perurbaions. This phenomenon is explained by he exisence of unsable small-ampliude oscillaions ha separae he uniform flow and he large-ampliude sop-and-go soluions. Along he exciable regime he ampliude of criical perurbaions decreases as he raffic becomes more and more dense unil he regime of sponaneous jam formaion is reached. A one end a vehicle mus almos sop o rigger a sop-and-go jam: acions have o be as large as he effecs. A he oher end here is sponaneous jam formaion: iny acions have large effecs. Tha is, he delayed car-following model esablishes a solid connecion beween he dynamics of coninuum models and he dynamics of non-delayed discree car-following models. Our resuls show ha in order o undersand he emergen behavior of raffic i is crucial o pay aenion o he behavior of individual drivers. A single driver can rigger a sopand-go wave by apping he brake hard enough. Doing so, he driver reduces he overall flux of he sysem significanly. Sill if such braking is below a criical limi, he ripples decay and he flow remains smooh. This behavior can only be deermined by sudying he inricae microscopic dynamics of he sysem and canno be concluded by simply looking a flux-densiy fundamenal diagrams 3,32. The obained dynamics should be aken ino accoun when opimizing ramp meering on highways 33, i.e., when he inflow from on-ramps are conrolled by raffic lighs based on flow measuremens. The curren algorihms only consider how much inflow is allowed in order o keep he densiy on he main highway below a cerain limi ha is given by he maximum of he fundamenal diagram. Our resuls vindicae ha i is also imporan ha vehicles reach heir desired speed before joining he main flow. This may be achieved by long enough on-ramps and leing only one or wo vehicles per green period ener he on-ramps. The diversiy of human drivers may sill impede our abiliy o conrol he emergen behavior of raffic on highways, since i is impossible o eliminae all irregulariies of driver behavior bad lane changes, sudden srong braking. However, many new vehicles are equipped wih Auonomous Cruise Conrol devices ha are able o measure disances beween vehicles and acuae he cars accordingly. Time delays arise in hese sysem due o he ime required for sensing, compuaion and acuaion. These are smaller han he human reacion ime bu sill need o be considered when designing he conrol algorihms 34,3. Furhermore, he effecs of uncerainies may also need o be considered in hese decenralized conrol sysems 36. ACKNOWLEDGMENTS One of he auhors G.O. acknowledges discussions wih Jeff Moehlis and Francesco Bullo. This research was suppored by he Insiue for Collaboraive Bioechnologies under Gran No. DAAD9-3-D4 from he U.S. Army Research Office, by he Hungarian Naional Science Foundaion under Gran No. OTKA K689, and by he EPSRC under Gran No. EP/E67/. 462-