This introductory paper reviews the current state-of-the-art scientific methods used for modelling, analysing and controlling the dynamics of vehicular traffic. Possible mechanisms underlying traffic jam formation and propagation are presented from a dynamical viewpoint. Stable and unstable motions are described that may give the skeleton of traffic dynamics, and the effects of driver behaviour are emphasized in determining the emergent state in a vehicular system. At appropriate points, references are provided to the papers published in the corresponding Theme Issue. Keywords: vehicular traffic; congestion; stop-and-go waves; Hopf bifurcation; driver reaction time; unstable waves I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, 'How many arbitrary parameters did you use for your calculations?' I thought for a moment about our cut-off procedures and said, 'Four'. He said, 'I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk '. (Dyson 2004, p. 297)  The introduction of the assembly line in the automotive industry about a century ago allowed the mass production of automobiles, which, in turn, revolutionized land transportation. At the same time, a problem was also generated that has not yet been resolved: traffic congestion. Since then, researchers from many different disciplines (mathematics, physics and engineering) have targeted this problem, *Author for correspondence (gabor@engineering.ucsb.edu). One contribution of 10 to a Theme Issue 'Traffic jams: dynamics and control'. This journal is © 2010 The Royal Society 4455 4456 G. Orosz et al. often using sophisticated mathematical tools brought from their own area of expertise. Also, analogies between traffic flow and other flows (fluid flow, gas flow and granular flow) were established. Although such analogies may help scientists to gain understanding of vehicular systems, it is becoming more and more obvious that traffic flows like no other flow in the Newtonian universe. To date, a vast number of different models have been constructed, but still no first principles have been established to guide the modelling procedure (if such principles exist at all). In many cases, authors claimed that the developed model described traffic better than models prior to that point, and such claims were often justified by fitting the models to empirical data. The quote at the beginning of this paper tries to illustrate, without questioning the importance of any specific model, that the above approach may easily lead to research capturing, but also missing, some essential characteristics. We believe that another way to conduct research in traffic can be by studying general classes of models and classifying their qualitative dynamical features (including 'hidden' unstable motions) when varying model parameters. To set such a dynamical viewpoint in the traffic community requires a unified mathematical language that allows scientists, working in different disciplines from mathematics through physics to engineering, to communicate. As a first step in this quest, we invited a group of world leading experts working on traffic problems in different fields to submit papers for this Theme Issue. Besides presenting novel results, they emphasize the fundamental dynamical features of traffic from their perspective. The papers published in this theme issue represent the state of the art of traffic science, and they are also aimed to be accessible to a broader audience. In this introductory paper (that also serves as an introduction to the theme issue), we provide readers with a brief overview of traffic data, modelling, dynamics and control. The presentation here is based on the simplest possible setting, i.e. vehicles following each other on a single lane, but we provide references to articles where such a concept has been extended for multilane traffic, junctions, traffic signals and urban traffic. Many of these extensions are also specifically targeted by one or more papers published in the theme issue. By the 1930s, vehicular traffic in the USA had already reached the level that required management. The first step towards this was to collect traffic data. Empirical data were collected by using photographic measurements at cross sections, and the data were summarized by the so-called flux-density or fundamental diagram (FD) where the flux (number of vehicles passing the cross section in unit time) was plotted as a function of the traffic density (Greenshields 1935) . Moreover, driver parameters, such as reaction time, were already recorded at that time (Greenshields 1936) . Since then, FDs have been used to characterize traffic behaviour. The FD in figure 1a, which is reproduced from Kerner (2004) , represents data collected from the German A5 autobahn near Frankfurt. Clearly, when constructing an FD, one 'projects' the behaviour of a very high-dimensional system to a very low-dimensional diagram. It is not clear, however, whether a state variable is plotted as a function of a parameter or a state space diagram is created; both interpretations are used in the literature. In recent decades, information technology has advanced substantially, and now it allows us to collect 'higher dimensional' traffic data. There are two different approaches used in practice. So-called macroscopic data are collected by installing sensors (usually double induction loop detectors) along the road that measure flux and speed at a certain location, and density may be calculated as the ratio of the two quantities. Usually the data are aggregated in time (e.g. in 1 min units) and used for generating spatio-temporal plots where the density and speed are colour-coded. Figure 1b shows such a diagram that is generated from data collected along the UK M25 motorway, which encircles London (Highways Agency, UK). When studying such a diagram, one may observe some distinctive spatio-temporal patterns. In certain regions, the velocity is almost homogeneous, that is, we say, uniform flow is observed. When the corresponding velocity is also high, it is often called free flow. In other regions the velocity changes in space and time; in particular, waves of congestion travel upstream with a characteristic speed of 15-20 km h −1 . One major goal in traffic research is to understand the mechanisms of formation and propagation of these so-called stop-and-go waves (also called wide moving jams sometimes). Indeed, for each detector, a separate FD may be constructed and analysed (Kerner 2004) . More recently, so-called microscopic data (describing the motion of individual vehicles) have become available. Such data are collected by mounting a camera on a place of higher elevation or on a helicopter from which individual vehicles can be traced and so vehicle trajectories may be reconstructed. In figure Macroscopic models that describe traffic in terms of density and velocity distributions (Eulerian description) can be fitted to macroscopic data, and model parameters can be estimated this way. Similarly, microscopic models that describe the behaviour of individual drivers (Lagrangian description) may be fitted to microscopic data, and the corresponding driver parameters can be determined. It is also possible to fit microscopic models to macroscopic data and vice versa, but this is less straightforward. Data-fitting for traffic systems is far from trivial, caused partly by the possibly large number of model parameters and partly by the fact that the studied high degree-of-freedom system changes in time; it is not that easy to fit an elephant in motion (see the quote at the beginning of this paper). In Wagner (2010), macroscopic data are used to estimate model parameters for both macroscopic and microscopic models, while, in Hoogendoorn & Hoogendoorn (2010) , microscopic models are fitted to microscopic data. In Laval & Leclercq (2010) , microscopic data are used to build microscopic models, and it has been proved essential to add dynamics to kinematic models to be able to reproduce traffic jam formation as observed in data. In this section, we give a brief review of the mathematics and physics literature about simplified traffic models; for more discussion, see Helbing (2001) and Nagel et al. (2003) . We do not attempt to give a complete list of models generated since the 1950s, but rather emphasize the dynamical assumptions behind the classes of models that have shaped traffic research in recent decades. We focus on models that are continuous in time and space, but also mention models where these quantities are discretized. Note that discretizing time and/or space in a multiscale system may 'wash away' some interesting dynamics and, consequently, may lead to limited or even incorrect conjectures. Phil. Trans. R. Soc. A (2010) Introduction. Traffic jams Depending on the spatial scales represented, two different classes of models exist. Macroscopic (or continuum) models do not consider individual vehicles but rather describe traffic in terms of continuous density and velocity distributions as functions of space and time. The dynamics, i.e. the time evolution of these distributions, is given by partial differential equations (PDEs). On the other hand, microscopic (or car-following) models describe the behaviour of individual vehicles: discrete entities that move in continuous space. Here, the motion of vehicles is given by ordinary and delay differential equations. Despite six decades of research, the connection between the two classes of models is still not clarified for two main reasons. On one hand, as mentioned above, the viewpoints of the two modelling approaches are different. The Eulerian view of continuum models deals with the changes of density and velocity in time at a fixed location, while, in the Lagrangian view of car-following models, distances and velocities are monitored when 'travelling together with individual vehicles'. Moreover, the relationship between distances and densities is far from trivial in dynamical situations (Berg et al. 2000) . On the other hand, the psychological actions of drivers make traffic different from any other flow. In this 'fluid', 'particles' (of finite size) interact by violating Newton's Third Axiom since the leader's motion influences the follower's motion but not vice versa-except when collisions occur. In continuum models, traffic is described via a density distribution r(x, t) and a velocity distribution v(x, t), which are continuous functions of location x and time t. In such models, we have conservation of vehicles, so that in the absence of sources and sinks (on-and off-ramps) where v stands for partial differentiation. The quantity q(x, t) := r(x, t)v(x, t) describes the instantaneous flux (also called throughput or flow) at location x and time t. Further refinements are possible considering multi-species models, multi-lane configurations and junctions (Daganzo 2002a,b; Benzoni-Gavage & Colombo 2003) . To complete the model, one needs to define the relationship between density and velocity (or alternatively between density and flux). One possibility is to assign the velocity directly as a function of density (and its spatial derivatives) Formulae (3.1), (3.2) are often called kinematic or first-order models. In the simplest setting, introduced in Lighthill & Whitham (1955) , we have where V(r) is a decreasing function, modelling that dense traffic should travel slower than sparse traffic for safety reasons. Equivalently, the flux can be assigned as q = rv = rV(r) := Q(r). If one initializes this model so that fast traffic is placed behind slow traffic, the solution profile sharpens and a discontinuous shock develops in finite time where the velocity jumps downwards, requiring infinite deceleration. To represent vehicles' inertia, dynamic or second-order models may be constructed where formula (3.1) is supplemented with an equation for the acceleration (i.e. the total derivative of the velocity) As a matter of fact, most models in the literature can be written in the specific form The first term on the right-hand side corresponds to relaxation to a densitydependent optimal velocity given by the decreasing function V(r) with a relaxation time T . The limit T → 0 suggests a connection with equation (3.3). Note that this idea also appears in microscopic models as explained further below. The second term is such that N = 0 for constant density and velocity, in which case equation (3.3) still holds. This term may be constructed using fluid dynamical analogies and usually contains a 'pressure term' that is proportional to −v x r/r and represents anticipation of drivers to events ahead, and a 'diffusion term' that is proportional to v xx and represents the averaged effects of noise (Payne 1979; Kerner & Konhäuser 1994) . Despite the fact that pressure and diffusion are abstractions that are difficult to interpret in traffic, these models are able to reproduce uniform flow as well as stopand-go waves (without discontinuities). However, they have been criticized for predicting non-physical flow motions (negative flux and negative velocities) under certain conditions (Daganzo 1995) . In response to this criticism N = rP (r)v x v has been proposed by Aw & Rascle (2000) , where the function P(r) represents anticipation. Moving N to the left-hand side and using equation (3.1), model (3.5) can be rewritten as Here v + P(r) represents a Lagrangian quantity and the model is claimed to properly describe the anisotropy of traffic flow: particles respond to stimuli from the front but not from the back. Notice that model (3.1), (3.6) is a hyperbolic PDE with a source term on the right-hand side. Continuum models can also be developed from microscopic models, yielding the form (3.5) where the spatial derivatives appear owing to the discrete-to-continuous transformation (Berg et al. 2000) . Time delays may also be included in continuum models (Siebel & Mauser 2006) to mimic driver reaction time. As will be shown below, reaction time is an essential element of car-following models, but it is not clear how it should be incorporated in continuum models. Noise may also be added explicitly to any of the above models, but it has rarely been done in the literature. The advantages of continuum models are that the results can be directly compared with macroscopic data, such as the FD in figure 1a and the spacetime diagram in figure 2b (Kerner 2004) , and that it is also relatively easy to provide the boundary conditions for velocity and flux. Their disadvantage is, however, that it is difficult to relate the model parameters to microscopic driver parameters. Moreover, it is not clear what limitations are imposed by the fact that fundamentally discrete systems are being modelled as continua. Introduction. Traffic jams Finally, we remark that there are Boltzman-type gas-kinetic models, which are based on collisions of particles and contain noise explicitly (Prigogine & Herman 1971; Helbing 1995) , but these cannot be presented in the form of (3.1), (3.2) or (3.1), (3.4). In car-following models, discrete entities move in continuous time and continuous space, as shown figure 2, where vehicles of equal length are plotted in a specific setting. At time t, the position of the front bumper of the nth car is denoted by x n (t), its velocity is v n (t) and the bumper-to-bumper distance to the vehicle in front (called the headway) is h n (t). Differentiating h n (t) = x n+1 (t) − x n (t) − with respect to time t and usingẋ n (t) = v n (t), one obtainṡ To complete the model this equation has to be supplemented with a car-following rule, that is, the velocity or the acceleration has to be given as the function of stimuli that are usually the distance h n , the velocity differenceḣ n and the vehicle's own velocity v n . Note that some models also contain the velocity v n+1 and/or the accelerationv n+1 of the leader, which will not be explicitly represented by the notation used below. To further simplify the matter, drivers with identical characteristics are often considered. When velocities are assigned, the corresponding kinematic models can be written in the form 8) which are sometimes called collision avoidance models (Kometani & Sasaki 1958) . At least one of the delays t, s, k has to be positive to obtain a feasible model, but it is not always obvious what these quantities physically mean. For example, in the model v n (t) = v n+1 (t − s), the delay s stands for a time gap (a time distance the driver keeps from the leader; Newell 2002), while in the model the delay t may represent a reaction time (a dead time required to process information and initiate action) and/or a relaxation time (that corresponds to the driver's reaction inertia; Newell 1961 Newell , 2002 Igarashi et al. 2001) . Here, V (h) is an increasing function that represents that the larger the headways are, the faster the drivers intend to go. To establish a connection between models (3.3) and (3.9), one may define V(r) := V (1/r − ). In more complicated kinematic models, like that in Gipps (1981) , usually the set-up t = s = k is used and the delay stands for the driver reaction time. Note that the kinematic rule (3.8) may require infinite accelerations. In dynamic models, the accelerations are prescribed aṡ Many other set-ups are also possible; for example, one may account for human memory effects by using distributed delays, as in Sipahi & Niculescu (2010) . Early dynamical car-following models, such as those in Gazis et al. (1961) , took the form 11) and used the human driver set-up; see Holland (1998) for a review. A shortcoming of these models is that they require no acceleration for zero relative velocity (ḣ = 0) independent of the distance h. That is, extremely small headways are allowed even when travelling with extremely high speed. A class of models that managed to eliminate this problem is called the optimal velocity model (OVM), The first term corresponds to relaxation to a density-dependent optimal velocity given by the increasing function V (h) with a relaxation time T . Clearly, for the human driver set-up, T = 0 leads to model (3.9). The increasing function W (ḣ) shows the dependence on the relative velocity. Notice that one may develop a continuum model like (3.5) from (3.12), but, even for W = 0, one obtains N = 0 since r = 1/(h + ) cannot be used in dynamical situations (Berg et al. 2000) . For W = 0, the model has been studied extensively without reaction time (Bando et al. 1995; Gasser et al. 2004) , using human driver set-up Bando et al. 1998; Orosz et al. 2010) . In this case, the model contains only a few parameters: the relaxation time T , the desired maximum velocity v max = max V (h) (attained for large h), the desired stopping distance h stop (below which V (h) ≡ 0) and one more parameter corresponding to a desired time gap (that is embedded in V ). Despite its simplicity, model (3.12) can reproduce qualitatively almost all kinds of traffic behaviour and also transitions between different behaviours. A shortcoming of the model is that some parameters (like the desired time gap) do not appear explicitly, while others (like the relaxation time T ) may be smaller than expected when fitting the model to data (Wagner 2010). Another car-following model that gained popularity in the last decade was introduced in Treiber et al. (2000) and is called the intelligent driver model (IDM), (3.13) In this model, all parameters have a clear physical meaning: a is the maximum acceleration, b is the comfortable deceleration, v max is the desired maximum velocity, h stop is the desired stopping distance and T gap is the desired time gap, and their values are as expected when fitting the model to data (Hoogendoorn & Hoogendoorn 2010). In some situations, the IDM produces unreasonably hard braking behaviour that has recently been eliminated by further adjustments (Kesting et al. 2010) . Numerous other dynamic models have been developed in the last two decades that are not listed here. We present only those that are simple in form and that we believe had a strong impact on traffic modelling. Note that much more complicated models may be implemented in simulation packages (MITSIM). There exist a couple of extensions of the car-following models (3.8) and (3.10). In multi-look-ahead models (Herman et al. 1959; Wilson et al. 2004 ) drivers respond to the motion of more than one vehicle ahead. Consequently, when monitoring M vehicles in front, the quantities . . , M , also appear in the governing equations (involving reaction times that are usually larger than t and s). Notice that, while using the variables h n , v n in systems (3.7), (3.8) and (3.7), (3.10), the vehicle length can be eliminated; this is not possible in the multi-lookahead case. Such next-nearest neighbour interactions can increase the (linear) stability of uniform flow. There also exist multi-species models (Mason & Woods 1997; Nagatani 2000) where drivers with different characteristics are considered, that is, g, f , , t, s, k → g n , f n , n , t n , s n , k n is assumed, which may be different for each n. However, usually only two different species are mixed, corresponding to automobiles and trucks. One might also extend car-following models to make them capable of modelling multi-lane traffic or junctions, in which case lane changes need to be handled (Davis 2004; Kesting et al. 2007) . Indeed, when a lane change occurs, the right-hand side of model (3.8) or (3.10) changes, that is, one needs to deal with a 'discontinuous dynamical system'. It is also possible to incorporate stochastic effects to model inconsistencies of driver behaviour and external disturbances (such as the weather or road unevenness). However, in the current literature, this is not typical, except in cases when the model is discretized in time. It has particular importance when space and velocities are also discretized (besides time), in which case we talk about cellular automata models (Nagel & Schreckenberg 1992) . These models used to be very popular because quick realizations of numerical simulations were possible, but the discretization may 'wash away' many dynamical effects that are important for understanding traffic dynamics. Before concluding this section, we collect the characteristic times appearing in traffic. We would like to emphasize that, even though these are close to each other in magnitude, their physical meaning and their effects on the dynamics may be completely different. -Reaction time. t, s, k ≈ 0.5-1.5 s is a dead time required to process information about stimuli and to initiate actions. It may be different for different kinds of stimuli; see model (3.10). -Relaxation time. T ≈ 1-10 s appears owing to the fact that vehicles are capable of only finite acceleration; see model (3.12). -Desired time gap (also called time headway). T gap ≈ 1-2 s is the time distance drivers would like to keep when following a leader; see model (3.13). We remark that these times should also be distinguished from the numerical update time that is simply an artefact of time discretization. However, depending on the discretization scheme, its effects may be similar to those of the multiples of the reaction time (Kesting & Treiber 2008) . In the following sections, all results are presented on the dynamic car-following model (3.7), (3.10), but most methods are also applicable to (3.7), (3.8) and (3.1), (3.4). As mentioned above, a traffic model should be able to reproduce uniform flow where both the velocities and the headways are time independent, that is, for identical vehicles, we have h n (t) ≡ h * ,ḣ n (t) ≡ 0 and v n (t) ≡ v * . (4.1) (For non-identical vehicles, the first formula would become h n (t) ≡ h * n , where h * n may be different for each vehicle.) We also assume that a functional relationship exists between the equilibrium headway h * and the equilibrium velocity v * , that is, 2) where V is assumed to have the following properties: -V is continuous and monotonously increasing (the more sparse traffic is, the faster drivers want to travel). -V (h) ≡ 0 for h ≤ h stop (in very dense traffic, drivers intend to stop). -V (h) ≡ v max for large h (in very sparse traffic, drivers intend to drive with maximum speed-often called free flow). Two examples are shown in figure 3a ,b. The function in figure 3a represents that between stopping and free-flow conditions-drivers intend to keep a constant time gap T gap . However, this function is non-smooth, which can complicate the Introduction. Traffic jams mathematical analysis of the corresponding models, so in many cases smoothed functions such as the one in figure 3b are used (that usually do not contain T gap explicitly). Such an equilibrium speed-density function has been explicitly built into the models (3.9) and (3.10), (3.12), where it is often called the optimal velocity function. For (3.10), (3.13), one finds that 4 and the corresponding V is similar in shape to figure 3b-except that V (h) < 0 for h ≤ h stop , i.e. cars intend to reverse in this regime. However, the uniform flow is usually unstable here, and such a nonphysical motion is rarely observed in simulations. Notice also that V (h stop ) = 1/T gap . Finally, we remark that the model (3.10), (3.11) does not admit such a uniform flow equilibrium: assuming v n (t) ≡ v * , any h n (t) ≡ h * n is possible (where h * n may be different for each vehicle, even for identical drivers). In equilibrium, one may define the equilibrium density and the flux as where V(r * ) = V (1/r * − ). This way, the equilibrium speed-headway diagrams in figure 3a,b can be transformed into the equilibrium flux-density (fundamental) diagrams in figure 3c,d . Indeed, the function V is explicitly built into the figure 3c is widely used in the literature because of its simplicity. In particular, when inserting it into the kinematic model (3.3), the fronts separating regions of different velocities (often called characteristics) propagate upstream with the constant speed c wave = −(h stop + )/T gap < 0. The equilibrium FDs nicely match the 'curve' on the left-hand side of the experimental diagram in figure 1a , but definitely do not do a good job describing the 'patch' on the right-hand side and the region where the curve and the patch overlap. To fix this shortcoming of equilibrium FDs, the non-equilibrium FDs in figure 3e,f are suggested in the literature (Kerner 2004)-in figure 3f , the whole shaded domain is part of the diagram. Clearly, dynamical effects need to be involved to explain these, especially if one wishes to explain the transitions between different states in the overlapping region. Some authors claim that such diagrams are the effect of a new 'phase' appearing in multi-lane traffic (Kerner 2009)-an explanation that has been criticized by others (Schönhof & Helbing 2009 ). In the next two sections, without trying to reconcile this conflict, we will show that, for sufficiently large reaction times, the dynamics of model (3.10) can reproduce figure 1a when considering only single-lane traffic. Furthermore, we will emphasize that it is useful to study 'hidden' unstable solutions (that are not observable in measurements) for characterizing the mechanisms of jam formation. There may be two obvious reasons why the right-hand side of the equilibrium FDs in figure 3c,d is not observable in empirical data. The equilibria either do not exist or they are not stable (solutions diverge and approach some other state). The latter case may be studied by linearizing model (3.10) about the equilibrium (equation (4.1) ). Defining the perturbationsh n (t) = h n (t) − h * ,ṽ n (t) = v n (t) − v * , one may obtain the linearized equationṡ where the derivatives are assumed to be positive to obtain physically realistic driver behaviour, i.e. drivers intend to decrease the perturbations. Indeed, we haveh n (t) =x n+1 (t) − There are (at least) three different methodologies to study the linear stability of the uniform flow (also often called string stability) that are used in the literature and lead to the same results. We named these approaches after the communities that use them the most. -'Physicist's approach'. An infinite road is considered and (without specifying the boundary conditions) the travelling wave ansatzx n (t) = Re(A e lt+iqn ), l ∈ C, q ∈ (0, 2p), is assumed. Indeed, this ansatz is equivalent to performing a Laplace transformation in time and a (discrete) Fourier transformation in space, and it leads to a second-order equation for l. To obtain stability, one needs to ensure Re(l) < 0 for all solutions (Bando et al. 1995; Wilson 2008 ). -'Dynamical systems approach'. A circular road configuration (i.e. periodic boundary conditions) is considered such that N vehicles are placed on a ring of length L + N (yielding h * = L/N and x N +1 (t) ≡ x 1 (t)). Usually, the large N limit is taken: N → ∞ such that L/N is kept constant. The trial solutionx n (t) = A n e lt , l, A n ∈ C, is assumed, which is equivalent to performing a Laplace transformation in time. This leads to a 2N th-order characteristic equation for l and again Re(l) < 0 ensures stability (Gasser et al. 2004; Orosz et al. 2010 ). -'Control engineer's approach'. A platoon configuration is considered such that the leader's motion can be assigned-in equilibrium, it travels with v * . Then the (n + 1)st and the nth vehicles are considered as an input (x n+1 ) output (x n ) system, and a transfer function is determined through a Laplace transformation. The condition for string stability requires the magnitude of the corresponding transfer function to be less than 1 for every excitation frequency (output has to be smaller than input) along the platoon (Ioannou & Xu 1994; Swaroop et al. 1994; Zhou & Peng 2005) . Notice that, in the second approach, the key parameter is the equilibrium headway h * , while, in the third approach, it is the equilibrium velocity v * . However, these quantities are linked by equation (4.2). The stability loss occurs for l = ±iu, and travelling waves with frequency u ∈ R + appear. In the dynamical system terminology, this kind of stability loss is called a Hopf bifurcation. One may calculate that A n = A e i(2kp/N )n , k = 1, . . . , N − 1, and so the small-amplitude waves can be written in the form where v amp = Au. This justifies the physicist's ansatz about the spatial patterns and gives q = 2kp/N , k = 1, . . . , N − 1. Note that k plays the role of a discrete wavenumber. In appendix A, the simple case of zero delays is considered. In this case, the characteristic equation has finitely many solutions for l, and it is easy to prove that all approaches give the same stability condition and that stability is lost to low-frequency, long-wavelength (k = 1) oscillations. For non-zero delays, however, the characteristic equation has infinitely many solutions for l owing to the terms e −lt , e −ls , e −lk . If the delays are sufficiently large, the flow may lose its stability to high-frequency, short-wavelength (k 1) oscillations (Kesting & Treiber 2008; Orosz et al. 2010) . Choosing even larger delays may lead to situations in which even the drivers' intentions to decrease the transients (expressed by F , G, H > 0 in equation (5.1)) fail to work and so the controller becomes unstable (Sipahi & Niculescu 2010) . Finally, we remark that the control engineer's approach may allow one to study convective instabilities (Pipes 1953; Ward & Wilson 2010) , while the dynamical systems approach is useful if one wishes to analyse the effects of nonlinearities. In the next section, we demonstrate that nonlinearities play an essential role in determining the emergent state of traffic systems.  The most popular way to study the behaviour of nonlinear traffic models is using numerical simulations. The great advantage of this approach is that almost any extension of model (3.10) can be implemented (non-identical drivers, multilane traffic, junctions). In some cases, it is more difficult to write the system in a closed mathematical form than implementing the algorithm for numerical simulations. Still, this approach has drawbacks. It may not be easy to determine the causes of certain behaviours appearing in simulations (especially in regions of multi-stability). This is due to the difficulty in detecting unstable solutions by numerical simulation in such a high-degree-of-freedom system without having some prior knowledge about them. As will be explained below, it is critical to explore such solutions if one wishes to understand the underlying mechanisms of jam formation. In this section, we consider a single lane with periodic boundary conditions, building the analysis on the dynamical systems approach outlined in the previous section. We show that the qualitative features of the empirical FD in figure 1a can be reproduced by a dynamic model (3.10) when considering finite reaction times. For more details about the techniques used and results presented, one may consult Gasser et al. (2004) , Orosz & Stépán (2006) and Orosz et al. (2009) . Some claims about multi-stability are also given in Krauss et al. (1997) , Lee et al. (1998) , Igarashi et al. (2001) , Safonov et al. (2002) and Helbing & Moussaid (2009) by using different tools. Here we consider the OVM (3.10), (3.12), but one may reproduce the analysisand obtain qualitatively similar behaviour-for any dynamic model that admits a uniform flow equilibrium (4.1), (4.2) with an equilibrium function that looks like figure 3b; for example, see the IDM (3.10), (3.13) except for small equilibrium headways. We consider the human driver set-up t = s > 0, k = 0 for the reaction times and assume the optimal and relative velocity functions The function V is shown in figure 3b . To decrease the number of parameters, we rescale the distances by the desired stopping distance h stop and rescale the velocities by the desired speed v max . However, to maintain the clarity of presentation, we do not represent this rescaling in the illustrations below. The corresponding dimensionless parameters are fixed as s/h stop = 2, /h stop = 1, bh stop /v max = 0.1, T v max /h stop = 2, tv max /h stop = 2. Thus, the only free parameter left in the model (3.10), (3.12), (6.1) is the equilibrium headway h * = L/N , and we show the qualitative changes in the dynamics when this parameter is varied. The results are given for N = 33, which is small enough to reveal the detailed microscopic dynamics but large enough to approximate the N → ∞ case well. First, the results of the linear stability investigations are shown. Considering the linearization (5.1), one obtains F = V (h * )/T , G = b, H = 1/T . The algebraic expressions for the stability boundaries are fairly complicated for t > 0, and they are not shown here. However, the stability criterion is qualitatively similar Phil. Trans. R. Soc. A (2010)
The theme issue on "Traffic jams: dynamics and control" REF highlights the growing interest in a dynamical systems viewpoint of transportation networks.
Traffic jams: dynamics and control
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Medicine Physics