tdgl and mkdv equations for jamming transition in the lattice models of traffic abstract the lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena they are the lattice versions of the hydrodynamic model of traffic two lattice models are presented one is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other is the difference equation in which both time and space variables are discrete we apply the linear stability theory and the nonlinear analysis to the lattice models it is shown that the time dependent ginzburg landau tdgl equation is derived to describe the traffic flow near the critical point a thermodynamic theory is formulated for describing the phase transitions and critical phenomena it is also shown that the perturbed modified korteweg de vries mkdv equation is derived to describe the traffic jam

time dependent ginzburg landau equation for the jamming transition in traffic flow abstract we present the thermodynamic theory describing the phase transition and critical phenomenon in traffic flow we derive the time dependent ginzburg landau tdgl equation through the modified korteweg de vries kdv equation from the car following model using the perturbation method we find the thermodynamic potential for the jamming transition it is shown that the coexisting and spinodal lines are obtained respectively from the first and second derivatives of the potential we prove that the jamming transition is the first order phase transition below the critical point and metastability exists between the coexisting and spinodal lines

computing covariant lyapunov vectors oseledets vectors and dichotomy projectors a comparative numerical study abstract covariant lyapunov vectors or oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations nonautonomous differentiable dynamical systems and random dynamical systems these vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles in recent years new computational methods for approximating oseledets vectors have been developed motivated by increasing model complexity and greater demands for accuracy in this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of ginelli et al 2007 36 and wolfe and samelson 2007 37 we compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry spectral separation and dimension we also investigate which methods perform well with limited data

perturbation theory for the approximation of stability spectra by qr methods for sequences of linear operators on a hilbert space abstract in this paper we develop a perturbation analysis for stability spectra lyapunov exponents and sacker sell spectrum for products of operators on a hilbert space both real and complex based upon the discrete qr technique error bounds are obtained in both the integrally separated and non integrally separated cases that correspond to distinct and multiple eigenvalues respectively for a single linear operator we illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation the time varying analogue of gaps between eigenvalues determines the sensitivity of the stability spectra to perturbation

empirical test for cellular automaton models of traffic flow based on a detailed microscopic test scenario motivated by recent empirical studies of single vehicle data several cellular automaton models for traffic flow are compared we find three levels of agreement with the empirical data 1 models that do not reproduce even qualitatively the most important empirical observations 2 models that are on a macroscopic level in reasonable agreement with the empirics and 3 models that reproduce the empirical data on a microscopic level as well our results are not only relevant for applications but also shed light on the relevant interactions in traffic flow

cellular automata simulating experimental properties of traffic flow a model for one dimensional traffic flow was developed using cellular automata the model showed the characteristic quantities of traffic flow and could be calibrated to empirical data the mechanism of traffic jam formation was deterministic rather than based on internal fluctuations the model was simple fast and could describe stop and go traffic and the characteristics of empirical highway traffic flow stable traffic flow at small traffic densities the formation of traffic jams above a certain traffic density and the movement of downstream congestion fronts

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

metastable states in a microscopic model of traffic flow it is a well known fact that metastable states of very high throughput and hysteresis effects exist in traffic flow which the simple cellular automaton model of traffic flow and its continuous generalization fail to reproduce it is shown that the model can be generalized to give a one parametric family of models a part of which reproduces the metastable states and the hysteresis the models that have that property and those that do not that are separated by a transition that can be clearly identified

computation of invariant tori by the method of characteristics in this paper we present a technique for the numerical approximation of a branch of invariant tori of finite dimensional ordinary differential equations systems our approach is a discrete version of the graph transform technique used in analytical work by fenichel indiana univ math j 21 1971 pp 193 226 in contrast to our previous work l dieci j lorenz and r d russell siam j sci statist comput 12 1991 pp 607 647 the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori but determines and updates such a coordinate system during a continuation process we give general convergence results for the method and present its algorithmic description we also show how the method performs on two physically important nonlinear problems a system of two coupled oscillators and the forced van der pol oscillator in the latter case we discuss some modifications needed to approximate an invariant curve for the poincare map

scaling behaviour in discrete traffic models we investigate the noisy discrete single lane highway traffic model proposed by nagel and schreckenberg 1992 and demonstrate by computer simulations that as a function of the car density there is a dynamical transition from laminar flow to jammed traffic in the system related to the divergence of the relaxation time of average car velocity the critical density is close to the position of the maximum in the fundamental diagram we give estimates of the critical exponents related to this jamming transition the critical density of the geometrical 1 1 dimensional percolation transition of jammed regions can be rather far from the jamming point depending on the strength of the noise which indicates that cooperativity leading to the jamming transition can be long range in character

jamming transition in two dimensional pedestrian traffic abstract jamming transitions of pedestrian traffic are investigated under the periodic boundary condition on the square lattice by the use of the lattice gas model of biased random walkers without the back step the two cases are presented the one with two types of walkers and the other with four types of walkers in the two types of walkers the first is the random walker going to the right and the second is the random walker going up in the four types of walkers the first second third and fourth are respectively the random walkers going to the right left up and down it is found that the dynamical jamming transitions occur at the critical densities the transition points do not depend on the system size for large system but depend strongly on the strength of drift the jamming transitions are compared with that obtained by the cellular automaton model of car traffic

towards a macroscopic modeling of the complexity in traffic flow based on the assumption of a safe velocity u e rho depending on the vehicle density rho a macroscopic model for traffic flow is presented that extends the model of the kuhne kerner konhauser by an interaction term containing the second derivative of u e rho we explore two qualitatively different forms of u e a conventional fermi type function and motivated by recent experimental findings a function that exhibits a plateau at intermediate densities i e in this density regime the exact distance to the car ahead is only of minor importance to solve the fluid like equations a lagrangian particle scheme is developed the suggested model shows a much richer dynamical behavior than the usual fluid like models a large variety of encountered effects is known from traffic observations many of which are usually assigned to the elusive state of synchronized flow furthermore the model displays alternating regimes of stability and instability at intermediate densities it can explain data scatter in the fundamental diagram and complicated jam patterns within this model a consistent interpretation of the emergence of very different traffic phenomena is offered they are determined by the velocity relaxation time i e the time needed to relax towards u e rho this relaxation time is a measure of the average acceleration capability and can be attributed to the composition e g the percentage of trucks of the traffic flow

exact solutions for a discrete system arising in traffic flow this paper concerns wave propagation in a discrete nonlinear system of equations proposed and studied by g f newell as a model for car following in traffic flow in particular newell found exact solutions for shock waves and related phenomena here exact solutions representing periodic waves and solitary waves are obtained the method relates travelling wave solutions to the toda and kac van moerbeke discrete systems in this and other ways much of the interest is in the general phenomena possible in discrete systems here including also a time lag rather than in just the specific traffic flow setting

modified kdv equation for jamming transition in the continuum models of traffic abstract continuum models of traffic are proposed to describe the jamming transition in traffic flow on a highway they are the simplified versions of the hydrodynamic model of traffic two continuum models are presented one is described by the partial differential equations and the other is the discrete lattice version the linear stability theory and the nonlinear analysis are applied to the continuum models the modified korteweg de vries equation kdv near the critical point is derived using the reduction perturbation method it is shown that the jamming transition and the density wave in the congested traffic flow are described by the modified kdv equation the solutions of the kdv equations obtained from the two models are compared with that of the optimal velocity model car following model

the density wave in a car following model according to the optimal velocity model the condition for stable traffic flow is deduced nonlinear analysis shows that the density fluctuation in traffic flow itself induces two types of local density waves a weak fluctuation occurring near the stability state in a wide range of headways forms a soliton determined by the korteweg de vries kdv equation the appearance of such a soliton shows that drivers tend to reach the safety distance when they are away from it this density wave degenerates to a travelling wave at the critical point a strong fluctuation occurring around the critical point forms a kink or a soliton determined by the modified korteweg de vries mkdv equation

asymptotic dynamics of self driven vehicles in a closed boundary we study the asymptotic dynamics of self driven vehicles in a loop using a car following model with the consideration of volume exclusions in particular we derive the dynamical steady states for the single cluster case and obtain the corresponding fundamental diagrams exhibiting two branches representative of entering and leaving the jam respectively by simulations we find that the speed average over all vehicles eventually reaches the same value regardless of final clustering states the autocorrelation functions for overall speed average and single vehicle speed are studied each revealing a unique time scale we also discuss the role of noises in vehicular accelerations based on our observations we give trial definitions about the degree of chaoticity for general self driven many body systems

differential games optimal control and directional derivatives of viscosity solutions of bellman s and isaacs equations recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two person zero sum differential games problems and the new notion of viscosity solutions of hamilton jacobi pde s introduced by m g crandall and p l lions in particular it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated hamilton jacobi bellman and isaacs equations in the present work it is shown that viscosity super and subsolutions of these equations must satisfy some inequalities called super and subdynamic programming principle respectively this is then used to prove the equivalence between the notion of viscosity solutions and the conditions introduced by a subbotin concerning the sign of certain generalized directional derivatives

phase transition in a difference equation model of traffic flow a difference equation is presented to describe traffic flow on a highway the difference equation model is derived from the optimal velocity models formulated in terms of the differential equations it is compared with the differential equation models we investigate phase transitions among the freely moving phase the coexisting phase in which jams appear and the uniform congested phase the linear stability theory is applied and the neutral stability line is obtained we find the critical point below which no jams appear to derive the modified korteweg de vries equation near the critical point we apply the reductive perturbation method we also compare the nonlinear analysis result with that of the optimal velocity model it is shown that the critical point and the amplitude of the jam are different from those of the optimal velocity model

remarks on hamilton jacobi equations with measurable time dependent hamiltonians on considere les equations de hamilton jacobi de la forme suivante u t h x t u u 0 dans r n 0 t ou h x t s p est continu en x s p et mesurable en t on considere une extension de la notion de solution viscosite au cas de h mesurable dependant du temps

jamming transitions and the modified korteweg de vries equation in a two lane traffic flow abstract the two lattice models are presented to simulate the traffic flow on a two lane highway they are the lattice versions of the hydrodynamic model of traffic the one model a is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other model b is the difference equation in which both time and space variables are discrete the jamming transitions among the freely moving phase the coexisting phase and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation the modified korteweg de vries mkdv equations are derived from the lattice models near the critical point the traffic jam is described by a kink antikink solution obtained from the mkdv equation it is found that the critical point the coexisting curve and the neutral stability line decrease with increasing the rate of lane changing also the computer simulation is performed for the model b it is shown that the coexisting curves obtained from the mkdv equation are consistent with the simulation result

the singular value decomposition to approximate spectra of dynamical systems theoretical aspects abstract in this paper we consider the singular value decomposition svd of a fundamental matrix solution in order to approximate the lyapunov and exponential dichotomy spectra of a given system one of our main results is to prove that svd techniques are sound approaches for systems with stable and distinct lyapunov exponents we also show how the information which emerges with the svd techniques can be used to obtain information on the growth directions associated to given spectral intervals

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

jamming transition in traffic flow on triangular lattice abstract phase transitions and critical phenomena are investigated in the two dimensional traffic flow on the triangular lattice numerically and analytically the two dimensional traffic model on the square lattice is extended to the traffic flow on the triangular lattice where the three roads cross on a site it is shown that the jamming transition between the freely moving and jammed phases depends on the configuration of car moving directions it is found that the three distinct jamming transitions occur the conventional jamming transition to the kink jams the jamming transition to the chaotic jams and the jamming transition to the oscillating jams the conventional jamming transition to the kink jams is analyzed by the use of the linear stability theory and the nonlinear method the coexisting curve between the freely moving and jammed phases is calculated from the solution of the modified korteweg de vries kdv equation

jamming transition of high dimensional traffic dynamics phase transition and critical phenomenon are investigated in high dimensional traffic flow numerically and analytically the two dimensional lattice traffic model is extended to the three and d dimensional traffic flows it is shown that the phase transition among the freely moving phase the coexisting phase and the uniformly congested phase occurs below the critical point in the two three and four dimensional traffic simulations the phase separation line and the critical point of the d dimensional traffic are consistent with those of the one dimensional traffic the time dependent ginzburg landau tdgl equation for the d dimensional traffic is derived by the use of the nonlinear analysis the phase separation line the spinodal line and the critical point are calculated from the tdgl equation it is found that the phase transition and critical phenomenon are independent of the space dimension

traffic behavior in a mixture of different vehicles the car following models are extended to take into account the characteristics of different vehicles the characteristic of each car can be included in the parameter of the dimensionless delay time the traffic behavior is investigated analytically and numerically under the periodic boundary condition at a low density all the vehicles move at a constant velocity when the density is larger than the critical value the traffic flow becomes unstable and the density waves appear the jamming transition does not depend only on the mean value of the dimensionless delay time but also on the variation with increase in the variation of dimensionless delay time the jamming transition occurs at a higher value of the mean value it is found that the variation of vehicles stabilizes the traffic flow and suppresses the traffic jams the transition line obtained by the linear stability analysis is compared with the simulation result

stability analysis of a multi phase car following model this paper presents a numerical stability analysis of a multi phase car following model under mild to severe disturbances the results show that local stability was always conformed an asymptotically unstable region was found for traffic in congested states one of the previously calibrated boundary conditions for close following situations was found to be in conflict with the stable condition required by the car following model which had attributed to speed oscillations during transition of the traffic from a non congested to a congested state suggestions were made to the choice of model parameter values to meet the stability conditions and ways to improve the model

computing sacker sell spectra in discrete time dynamical systems in this paper we develop boundary value methods for detecting sacker sell spectra in discrete time dynamical systems the algorithms are advancements of earlier methods for computing projectors of exponential dichotomies the first method is based on the projector residual p 2 p if this residual is large then the difference equation has no exponential dichotomy further criterions for detecting sacker sell spectral intervals are the norm of end points and midpoints of the solution of a specific boundary value problem refined error estimates for the underlying approximation process are given and the resulting algorithms are applied to an example with known continuous sacker sell spectrum as well as to the variational equation along orbits of henon s map

viscosity solutions of isaacs equations and differential games with lipschitz controls abstract it is demonstrated that the upper and lower values of a two person zero sum differential game solve the respective upper and lower isaacs equations in the viscosity sense introduced by crandall and lions trans amer math soc 277 1983 1 42 since such solutions are unique this yields a fairly simple proof that the game has value should the minimax condition hold as a further application of viscosity techniques a new and simpler proof that the upper and lower values can be approximated by the values of certain games with lipschitz controls is given

cellular automaton model within the fundamental diagram approach reproducing some findings of the three phase theory abstract we propose a simple cellular automaton for traffic flow within the fundamental diagram which could reproduce aspects of the three phase theory this so called average space gap model asgm is based on the nagel schreckenberg model with additional slow to start and anticipation rules the anticipation rule takes into account the average space gap of multiple leading vehicles and conveys to the model its three phase property due to the anticipation rule asgm can show the transition from free flow to synchronized flow due to the slow to start rule asgm can show the spontaneous wide moving jam emerges in the synchronized flow simulations are carried out for periodic and open boundary conditions under periodic boundary condition the fundamental diagram and the properties of synchronized flow are studied under open boundary condition different congested patterns induced by an on ramp are analyzed we found that the asgm produces the same spatiotemporal dynamics as many of the more complex three phase models due to its simplicity and its close relation to conventional slow to start models this model can shed light on the relation between two phase and three phase models

discrete stochastic models for traffic flow we investigate a probabilistic cellular automaton model which has been introduced recently this model describes single lane traffic flow on a ring and generalizes the asymmetric exclusion process models we study the equilibrium properties and calculate the so called fundamental diagrams flow versus density for parallel dynamics this is done numerically by computer simulations of the model and by means of an improved mean field approximation which takes into account short range correlations for cars with a maximum velocity of 1 the simplest nontrivial approximation gives the exact result for higher velocities the analytical results obtained by iterated application of the approximation scheme are in excellent agreement with the numerical simulations

traffic jamming on hexagonal lattice abstract a traffic model on hexagonal lattice is studied the moving objects are modeled as biased random walkers on hexagonal lattice the hexagonal lattice traffic model is an extension of the square lattice traffic model proposed by muramatsu et al physica a 267 1999 487 the hexagonal and square models are compared it is found that hexagonal and square models have a phase transition from freely moving state to jammed state at a critical density the critical density of hexagonal model is greater than that of square model when the drift of moving objects is high the difference between the critical densities of hexagonal and square models decreases when the drift decreases

a perturbation theorem for exponential dichotomies suppose the linear equation x a t x has an exponential dichotomy and suppose b t is close to a t in the following sense on any interval of length 2t b t is close to some translate a t of a t actually the conditions in the paper are slightly weaker than this then if t is sufficiently large the equation y b t y also has an exponential dichotomy this generalises the usual roughness theorem for exponential dichotomies

traffic model with an absorbing state phase transition we consider a modified nagel schreckenberg ns model in which drivers do not decelerate if their speed is smaller than the headway number of empty sites to the car ahead in the original ns model such a reduction in speed occurs with probability p independent of the headway as long as the current speed is greater than zero in the modified model the free flow state with all vehicles traveling at the maximum speed v max is absorbing for densities smaller than a critical value c 1 v max 2 the phase diagram in the p plane is reentrant for densities in the range c c both small and large values of p favor free flow while for intermediate values a nonzero fraction of vehicles have speeds v max in addition to representing a more realistic description of driving behavior this change leads to a better understanding of the phase transition in the original model our results suggest an unexpected connection between traffic models and stochastic sandpiles

lyapunov spectral intervals theory and computation different definitions of spectra have been proposed over the years to characterize the asymptotic behavior of nonautonomous linear systems here we consider the spectrum based on exponential dichotomy of sacker and sell j differential equations 7 1978 pp 320 358 and the spectrum defined in terms of upper and lower lyapunov exponents a main goal of ours is to understand to what extent these spectra are computable by using an orthogonal change of variables transforming the system to upper triangular form and the assumption of integral separation for the diagonal of the new triangular system we justify how popular numerical methods the so called continuous qr and svd approaches can be used to approximate these spectra we further discuss how to verify the property of integral separation and hence how to a posteriori infer stability of the attained spectral information finally we discuss the algorithms we have used to approximate the lyapunov and sacker sell spectra and present some numerical results

exponential separation exponential dichotomy and spectral theory for linear systems of ordinary differential equations abstract this paper is concerned with linear nonautonomous systems of ordinary differential equations a criterion for exponential separation in terms of exponential dichotomy is given as corollaries we obtain the roughness theorem for exponential separation and the new result that an upper triangular system on a half line is exponentially separated if and only if the system corresponding to its diagonal is a minimal decomposition into exponentially separated subspaces is defined it turns out that it is in general finer than the sacker sell spectral decomposition but that the two decompositions coincide for almost periodic systems

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

exponential dichotomy on the real line svd and qr methods abstract in this work we show when and how techniques based on the singular value decomposition svd and the qr decomposition of a fundamental matrix solution can be used to infer if a system enjoys or not exponential dichotomy on the whole real line

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

delay effect on phase transitions in traffic dynamics a dynamical model of traffic is proposed to take into account the effect of acceleration delay in the limit of no delay the model reproduces the optimal velocity model of traffic when the delay is small it is shown that the phase transition among the freely moving phase the coexisting phase and the uniform congested phase occurs below the critical point above the critical point no phase transition occurs the value a c of the critical point increases with increasing delay time 1 b where a is the friction coefficient or sensitivity parameter when the delay time is longer than frac 1 2 the critical point disappears and the phase transition always occurs the linear stability theory and nonlinear analysis are applied the critical point predicted by the linear stability theory agrees with the simulation result the modified korteweg de vries kdv equation is obtained from the nonlinear analysis near the critical point the phase separation line obtained from the modified kdv equation is consistent with the simulation result

jamming transition in pedestrian counter flow a lattice gas model with biased random walkers is presented to mimic the pedestrian counter flow in a channel under the open boundary condition of constant density there are two types of walkers without the back step the one is the random walker going to the right and the other is the random walker going to the left it is found that a dynamical jamming transition from the freely moving state at low density to the stopped state at high density occurs at the critical density the transition point is given by pc 0 45 0 01 not depending on the system size the transition point depends on the strength of drift and decreases with increasing drift also we present the extended model to take into account the traffic rule in which a pedestrian walks preferably on the right hand side of the channel

gas kinetic derivation of navier stokes like traffic equations macroscopic traffic models have recently been severely criticized to base on lax analogies only and to have a number of deficiencies therefore this paper shows how to construct a logically consistent fluid dynamic traffic model from basic laws for the acceleration and interaction of vehicles these considerations lead to the gas kinetic traffic equation of paveri fontana its stationary and spatially homogeneous solution implies equilibrium relations for the fundamental diagram the variance density relation and other quantities which are partly difficult to determine empirically paveri fontana s traffic equation allows the derivation of macroscopic moment equations which build a system of non closed equations this system can be closed by the well proved method of chapman and enskog which leads to euler like traffic equations in zeroth order approximation and to navier stokes like traffic equations in first order approximation the latter are finally corrected for the finite space requirements of vehicles it is shown that the resulting model is able to withstand the above mentioned criticism