Abstract and Applied Analysis

Volume 2015, Article ID 167430, 10 pages

http://dx.doi.org/10.1155/2015/167430

## Synchronization Transition and Traffic Congestion in One-Dimensional Traffic Model

^{1}School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China^{2}Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022, China

Received 23 June 2014; Accepted 28 August 2014

Academic Editor: Yongli Song

Copyright © 2015 Zhi Xin and Jian Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A nonlinear car-following model with driver’s reaction time is studied from the synchronization transition viewpoint. We investigate the traffic congestion from the view of chaos system synchronization transition. Our result shows that the uniform flow corresponds to the complete synchronization and the stop-and-go congested state corresponds to the lag synchronization of the vehicles. An analytical criterion for synchronization manifolds stability is obtained; the analytical result and the numerical result are consistent. The synchronization transition is also trigged by the driver’s reaction time. We analyze the car-following model by the use of the nonlinear analysis method and derive the modified KdV equation describing the kink density wave.

#### 1. Introduction

Traffic jams are very annoying in our life and have been studied by many physicists [1]; however, the precise mechanism for generation and propagation of traffic jams is still not clearly understood. Some traffic models have been established to study the dynamics of traffic flow, such as car-following models, cellular automaton models, gas kinetic models, and hydrodynamic models [2].

Car-following models or microscopic models describe the behaviors of individual vehicles, which are described by ordinary differential equation or delay differential equation. Recently, one more widely used car-following model is the optimal velocity model [3, 4]; by this kind of model, the effects of fluctuations of traffic jams are analyzed, the jamming transitions and density waves have been invested [5–9], and the bifurcation phenomena of the oscillating solution are also explored by using numerical continuations techniques [10–12]. Another popular car-following model is the intelligent driver model that was introduced by Treiber et al. [13, 14]; in this model, all parameters have a clear physical meaning; congested traffic states have been empirically observed and microscopically simulated. Many models are able to explain uniform flow as well as stop-and-go waves; however, the transition [15–28] between the two qualitatively different solutions is still not clarified.

According to the car-following model theory, for each individual vehicle, an equation of motion is the analogue of Newton’s equation for each individual particle in a system of interacting classical particles. In Newtonian mechanics, the acceleration may be regarded as the response of the particle to the stimulus, and it receives in the form of force which includes both the external force and those arising from its interaction with the other particles in the system. In traffic system, each driver can respond to the surrounding traffic conditions only by accelerating or decelerating the vehicle; so, the basic philosophy of the car-following theories can be summarized by the equation for the th vehicle ().

We know, for a chaotic system, the appearances and robustness of chaotic synchronization states have been established by means of different coupling schemes [29], one of which is Pecora and Carroll method (unidirectional coupling or drive-response coupling).

In this paper, we regard the car-following system as a drive-response coupling chaotic system; we make use of chaotic systems synchronization transition [30–34] method to study the synchronization transition of microscopic movement of the vehicles and further reveal the relationship between synchronization transition and traffic congestion. First, we use the long wave expansion method to give an analytical criterion for synchronization manifolds stability. In order to verify the analytical result, we use DDE-BIFTOOL to perform a two-parameter bifurcation analysis of the model; due to the demands on the CPU time and the memory, the investigation was restricted to the setting of vehicles, and the analytical result is consistent with the numerical result. Second, we find different transition region in two-parameter plane and the vehicles display different motion. Third, we consider how the driver’s reaction time impacts on the synchronization transition. Finally, we analyze the car-following model by the use of the nonlinear analysis method and we derive the modified KdV equation describing the kink density wave.

The layout of this paper is organized as follows. We introduce the car-following model in Section 2 and derive analytical criteria for synchronization manifolds stability in Section 3. Numerical simulation is in Section 4. Nonlinear analysis is in Section 5. In Section 6, we present the main results.

#### 2. Car-Following Model

Here, we consider a single lane of traffic with identical vehicles; displacements and velocities are denoted as and , respectively; the spacing of adjacent vehicle is called headway (). For the sake of simplicity, we suppose that vehicles are placed on a circular road of length (). The headway consists of the condition The acceleration of the th vehicle is given by The is called the sensitivity of the vehicles and the is driver’s reaction time. The function is optimal velocity function; it has the following properties.(a) is a nonnegative, continuous, and monotone increasing function.(b) as .(c)There exists a jam headway such that for .

The dimensional parameter OV function [12] is We introduce the rescaled variables , ; the OV function becomes Model (1) and model (2) are transformed into Here , ; in the remainder of this paper we take the rescaled OV function and model. In order to express simplicity, the OV function and model are expressed:

#### 3. Synchronization Transition

The synchronization manifold of the system (7) and (8) is The system ((7), (8)) possesses uniform flow equilibrium All vehicles reach complete synchronization. The stability of the synchronization manifolds will change when the parameter is varied. To see whether the synchronization manifold of the system ((7), (8)) is stable or not, we add a small perturbation

According to (7) and (8), we can calculate a linearized equation with respect to the uniform flow (10). The linearized equation is

Indeed, we have , , and .

Orosz et al. have used the dynamical system approach and numerical continuation technique to give out the stable curves, but the numerical continuation method (DDE-BIFTOOL) needs a lot of CPU time and the memory; below, we can find a simple method to analyze the linear stability of the synchronization manifold [10].

*Physical Approach*. Let , , ; the characteristic equation is given by
We know that the leading term of is of order . When , . We can derive the long wave expansion of , which is determined order by order around . By expanding
the first- and second-order terms of are obtained:
If is a positive value, the synchronization manifold is stable; if is a negative value, the synchronization manifold is unstable.

By , we get Equation (16) is the analytical criterion for synchronization manifold stability.

If the number of the vehicles is large, to obtain the stable boundary curve by the numerical continuations method requires a lot of time. In order to compare with numerical method, we now focus on the case of vehicles. For , the stable boundary curve is shown in Figure 1. We use the numerical continuation techniques (DDEBIFTOOL) to get the blue curve and we get the analytical curve (red curve) according to (16). We can find that the blue curve and the red curve completely overlap, and this shows that our analytical approach is correct. In Section 4 numerical simulations also verify our conclusion.