Global bifurcation investigation of an optimal velocity traffic model with driver reaction time

Bifurcation control and analysis of traffic flow model based on driver prediction effect

Based on the measured data, in this paper, we find that there are significant differences in the ideal speed of drivers with different attributes during their driving process. Through wavelet analysis, it is proved that the regularity of the car-following behavior of the driver is poor in a short time, and it cannot be predicted and analyzed at a small time scale. As one of the important participants in the traffic system, it is necessary to consider the driving behavior factors in the traffic flow model. Therefore, in this paper, we propose a nonuniform continuous traffic flow model. Based on the difference in the expected headway of the driver, the model considers the self-stabilization effect of the prediction time of the driver on the optimal speed difference and considers control theory to further improve the model. Through this model, the bifurcation theory can be applied to the stability analysis of the traffic system to study the stability mutation behavior of the traffic system at the bifurcation point. Through linear and nonlinear analysis methods, the stability conditions of the model can be derived, and the type of equilibrium point of the model can be judged. In addition, the random function is applied to the traffic model, and the bifurcation control is carried out for the bifurcation behavior in the traffic; that is, the bifurcation characteristics of the traffic system are changed by designing linear and nonlinear random feedback controllers. In this paper, we theoretically prove the existence conditions and types of Hopf bifurcation at the equilibrium point of the model and analyze the internal causes of the stability mutation of the traffic flow through the Hopf bifurcation point. Then a feedback controller is designed to control the amplitude of the Hopf bifurcation and the limit cycle formed by the Hopf bifurcation. Finally, the theoretical derivation results are verified by experimental numerical simulation. In this paper, we show that, by adjusting the control parameters of the feedback controller, the Hopf bifurcation of the stochastic system can be delayed or even eliminated, and the amplitude of the limit cycle formed by the Hopf bifurcation can be adjusted to achieve the purpose of controlling the stability of the traffic system and prevent or alleviate traffic congestion.

Review of sample-based methods used in an analysis of multistable dynamical systems

Sample-based methods are a useful tool in analyzing the global behavior of multi-stable systems originating from various branches of science. Classical methods, such as bifurcation diagrams, Lyapunov exponents, and basins of attraction, often fail to analyze complex systems with many coexisting attractors. Thus, we have to apply a different strategy to understand the dynamics of such systems. We can distinguish basin stability, extended basin stability, constrained basin stability, basin entropy, time dependent stability margin, and survivability among sample-based methods. Each method has specific properties and gives us important data about the behavior of the analyzed system. However, none of the methods provides complete information. Hence, to have a full overview of the dynamics, one has to collect data from two or more approaches. This study describes the sample-based methods and presents their advantages and disadvantages for the archetypal nonlinear oscillator with multiple coexisting attractors. Hence, we give helpful information in selecting the best method or methods for analyzing the dynamical system.

A Vehicle Guidance Model with a Close-to-Reality Driver Model and Different Levels of Vehicle Automation

This paper presents a microscopic vehicle guidance model which adapts to different levels of vehicle automation. Independent of the vehicle, the driver model built is different from the common microscopic simulation models that regard the driver and the vehicle as a unit. The term “Vehicle Guidance Model” covers, here, both the human driver as well as a combination of human driver and driver assistance system up to fully autonomously operated vehicles without a (human) driver. Therefore, the vehicle guidance model can be combined with different kinds of vehicle models. As a result, the combination of different types of driver (human/machine) and different types of vehicle (internal combustion engine/electric) can be simulated. Mainly two parts constitute the vehicle guidance model in this paper: the first part is a traditional microscopic car-following model adjusted according to different degrees of automation level. The adjusted model represents the automation level for the present and the near and the more distant future. The second part is a fuzzy control model that describes how humans adjust the pedal position when they want to reach a target speed with their vehicle. An experiment with 34 subjects was carried out with a driving simulator based on the experimental data and the fuzzy control strategy was determined. Finally, when comparing the simulated model data and actual driving data, it is found that the fuzzy model for the human driver can reproduce the behavior of human participants almost accurately.

Basins of attraction of the bistable region of time-delayed cutting dynamics

This paper investigates the effects of bistability in a nonsmooth time-delayed dynamical system, which is often manifested in science and engineering. Previous studies on cutting dynamics have demonstrated persistent coexistence of chatter and chatter-free responses in a bistable region located in the linearly stable zone. As there is no widely accepted definition of basins of attraction for time-delayed systems, bistable regions are coined as unsafe zones (UZs). Hence, we have attempted to define the basins of attraction and stability basins for a typical delayed system to get insight into the bistability in systems with time delays. Special attention was paid to the influences of delayed initial conditions, starting points, and states at time zero on the long-term dynamics of time-delayed systems. By using this concept, it has been confirmed that the chatter is prone to occur when the waviness frequency in the workpiece surface coincides with the effective natural frequency of the cutting process. Further investigations unveil a thin "boundary layer" inside the UZ in the immediate vicinity of the stability boundary, in which we observe an extremely fast growth of the chatter basin stability. The results reveal that the system is more stable when the initial cutting depth is smaller. The physics of the tool deflection at the instant of the tool-workpiece engagement is used to evaluate the cutting safety, and the safe level could be zero when the geometry of tool engagement is unfavorable. Finally, the basins of attraction are used to quench the chatter by a single strike, where the resultant "islands" offer an opportunity to suppress the chatter even when the cutting is very close to the stability boundary.

Hopf bifurcation analysis for a dissipative system with asymmetric interaction: Analytical explanation of a specific property of highway traffic

A dissipative system with asymmetric interaction, the optimal velocity model, shows a Hopf bifurcation concerned with the transition from a homogeneous motion to the formation of a moving cluster, such as the emergence of a traffic jam. We investigate the properties of Hopf bifurcation depending on the particle density, using the dynamical system for the traveling cluster solution of the continuum system derived from the original discrete system of particles. The Hopf bifurcation is revealed as a subcritical one, and the property explains well the specific phenomena in highway traffic: the metastability of jamming transition and the hysteresis effect in the relation of car density and flow rate.

Stochastic switching in delay-coupled oscillators

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators.

Linear stability analysis of first-order delayed car-following models on a ring

The evolution of a line of vehicles on a ring is modeled by means of first-order car-following models. Three generic models describe the speed of a vehicle as a function of the spacing ahead and the speed of the predecessor. The first model is a basic one with no delay. The second is a delayed car-following model with a strictly positive parameter for the driver and vehicle reaction time. The last model includes a reaction time parameter with an anticipation process by which the delayed position of the predecessor is estimated. Explicit conditions for the linear stability of homogeneous configurations are calculated for each model. Two methods of calculus are compared: an exact one via Hopf bifurcations and an approximation by second-order models. The conditions describe stable areas for the parameters of the models that we interpret. The results notably show that the impact of the reaction time on the stability can be palliated by the anticipation process.

Car-following model with relative-velocity effect and its experimental verification

In driving a vehicle, drivers respond to the changes of both the headway and the relative velocity to the vehicle in front. In this paper a new car-following model including these maneuvers is proposed. The acceleration of the model becomes infinite (has a singularity) when the distance between two vehicles is zero, and the asymmetry between the acceleration and the deceleration is incorporated in a nonlinear way. The model is simple but contains enough features of driving for reproducing real vehicle traffic. From the linear stability analysis, we confirm that the model shows the metastable homogeneous flow around the critical density, beyond which a traffic jam emerges. Moreover, we perform experiments to verify this model. From the data it is shown that the acceleration of a vehicle has a positive correlation with the relative velocity.

Fluid-dynamical and microscopic description of traffic flow: a data-driven comparison

Much work has been done to compare traffic-flow models with reality; so far, this has been done separately for microscopic, as well as for fluid-dynamical, models of traffic flow. This paper compares directly the performance of both types of models to real data. The results indicate that microscopic models, on average, seem to have a tiny advantage over fluid-dynamical models; however, one may admit that for most applications, the differences between the two are small. Furthermore, the relaxation times of the fluid-dynamical models turns out to be fairly small, of the order of 2 s, and are comparable with the results for the microscopic models. This indicates that the second-order terms are weak; however, the calibration results indicate that the speed equation is, in fact, important and improves the calibration results of the models.

Dynamical phenomena induced by bottleneck

We study a microscopic follow-the-leader model on a circle of length L with a bottleneck. Allowing large bottleneck strengths we encounter very interesting traffic dynamics. Different types of waves--travelling and standing waves and combinations of both wave types--are observed. The way to find these phenomena requires a good understanding of the complex dynamics of the underlying (nonlinear) equations. Some of the phenomena, like the ponies-on-a-merry-go-round solutions, are mathematically well known from completely different applications. Mathematically speaking we use Poincaré maps, bifurcation analysis and continuation methods beside numerical simulations.

Stability of car following with human memory effects and automatic headway compensation

This paper addresses the study of some appropriate control strategies in order to guarantee the exponential stability of a class of deterministic microscopic car-following models including human drivers' memory effects and automated headway controllers. More precisely, the delayed action/decision of human drivers is represented using distributed delays with a gap and the considered automated controller is of proportional derivative type. The analysis is performed in both delay parameter and controller gain parameter spaces, and appropriate algorithms are proposed. Surprisingly, large delays and/or gains improve stability for the corresponding closed-loop schemes. Finally, some illustrative examples as well as various interpretations of the results complete the presentation.

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

A nonlinear car-following model is studied with driver reaction time delay by using state-of-the-art numerical continuations techniques. These allow us to unveil the detailed microscopic dynamics as well as to extract macroscopic properties of traffic flow. Parameter domains are determined where the uniform flow equilibrium is stable but sufficiently large excitations may trigger traffic jams. This behavior becomes more robust as the reaction time delay is increased.

Dissipative system with asymmetric interaction and Hopf bifurcation

A dissipative system with asymmetric interaction, as well as the optimal velocity model, generally shows a Hopf bifurcation concerned with the transition from homogeneous motion to the formation of nontrivial patterns. We reveal that the origin of Hopf bifurcation in macroscopic phenomena is strongly related to asymmetric interaction in a microscopic many-body system, using the continuum system derived from the original discrete system.

Mechanisms for spatio-temporal pattern formation in highway traffic models

A key qualitative requirement for highway traffic models is the ability to replicate a type of traffic jam popularly referred to as a phantom jam, shock wave or stop-and-go wave. Despite over 50 years of modelling, the precise mechanisms for the generation and propagation of stop-and-go waves and the associated spatio-temporal patterns are in dispute. However, the increasing availability of empirical datasets, such as those collected from motorway incident detection and automatic signalling system (MIDAS) inductance loops in the UK or the next-generation simulation trajectory data (NGSIM) project in the USA, means that we can expect to resolve these questions definitively in the next few years. This paper will survey the essence of the competing explanations of highway traffic pattern formation and introduce and analyse a new mechanism, based on dynamical systems theory and bistability, which can help resolve the conflict.

A vehicle overtaking model of traffic dynamics

Mathematical models that describe the dynamical behavior of a group of vehicles as they move along a stretch of road are known as car following models. They attempt to model the interactions between individual vehicles where the behavior of each vehicle is dependent on the motion of the vehicle directly in front and overtaking is not permitted. In this paper, the traditional car following model is modified by removing this "no overtaking" restriction and its behavior is investigated for a group of vehicles traveling on a closed loop. The resulting model is described in terms of a set of coupled time delay differential equations, and these are solved numerically to analyze their post transient behavior under a periodic perturbation. The effect of varying both the time taken for the driver to respond to the behavior of the vehicle in front and the length of the closed loop is examined. For certain parameter choices, the post transient behavior is found to be chaotic, and in these cases the degree of chaos is estimated using the Grassberger-Procaccia dimension.

Hysteresis phenomena of the intelligent driver model for traffic flow

We present hysteresis phenomena of the intelligent driver model for traffic flow in a circular one-lane roadway. We show that the microscopic structure of traffic flow is dependent on its initial state by plotting the fraction of congested vehicles over the density, which shows a typical hysteresis loop, and by investigating the trajectories of vehicles on the velocity-over-headway plane. We find that the trajectories of vehicles on the velocity-over-headway plane, which usually show a hysteresis loop, include multiple loops. We also point out the relations between these hysteresis loops and the congested jams or high-density clusters in traffic flow.

Subcritical Hopf bifurcations in a car-following model with reaction-time delay

A nonlinear car-following model of highway traffic is considered, which includes the reaction-time delay of drivers. Linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via Hopf bifurcations and thus oscillations can appear. The stability and amplitudes of the oscillations are determined with the help of normal-form calculations of the Hopf bifurcation that also handles the essential translational symmetry of the system. We show that the subcritical case of the Hopf bifurcation occurs robustly, which indicates the possibility of bistability. We also show how these oscillations lead to spatial wave formation as can be observed in real-world traffic flows.

Stability of the car-following model on two lanes

In the case of two-lane traffic, vehicle drivers always worry about the lane changing actions from neighbor lane. This paper studies the stability of a car-following model on two lanes which incorporates the lateral effects in traffic. The stability condition of the model is obtained by using the linear stability theory. The modified Korteweg-de Vries equation is constructed and solved, and three types of traffic flows in the headway-sensitivity space--stable, metastable, and unstable--are classified. Both analytical and simulation results show that the anxiousness about lane changing from neighbor lane indeed has influence upon people's driving behavior and the consideration of lateral effects could stabilize the traffic flows on both lanes.