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In this study, the influence of traveler's departure time choice in day-to-day dynamic evolution of traffic flow in a transportation network is investigated. Combining historical information and real-time information, a dynamic evolution model of traffic flow with a study period divided into two intervals is proposed for a simple two-link network. Then, the evolution of network traffic flow is investigated using numerical experiments. Three types of information are considered:

Both traditional static and dynamic traffic assignments only focus on the equilibrium solution and its solution algorithm. The research on the evolution of network traffic flow focuses on finding whether the equilibrium of network flows exists and has been extensively analyzed. Such research does not focus on network flows equilibrium but on the dynamic evolution process of network traffic flow, exploring whether equilibrium exists in network flows and how the equilibrium is reached.

The dynamic evolution of network traffic flow is the macroscopic outcome of a large number of traveler’s route choices. Therefore, from the view of individual travelers, different assumptions about traveler’s choice behavior may lead to different results of network traffic-flow evolution. Nakayama et al. [

The preceding research was carried out using microscopic simulation. However, some scholars have simulated day-to-day route choice behavior at microlevel using experimental methods. Avineri et al. [

On the other hand, other scholars studied the evolution of network traffic flow directly from aggregate-perspective. Smith [

The evolution of network traffic flow is a discrete dynamic system that has attracted the interest of many researchers who used nonlinear dynamics to analyze this phenomenon. Cantarella et al. [

The preceding research on the evolution of network traffic flow only considers the influence of route choice, but does not consider the influence of departure time choice. It assumes that traveler’s route choice is based on a single departure interval. However, traveler’s departure time choice behavior is also an important factor that influences the evolution of network traffic flow. It is reasonable to relax this assumption because travelers will travel at different departure times to reduce their travel times. Therefore, it is necessary to analyze the evolution of network traffic flow under the combined influence of traveler’s departure time and route choice behavior. The departure time is not fixed, and travelers adjust the departure time according to the congestion level of the network. Based on this concept, Cascetta and Cantarella [

Note that the research previously conducted by the authors to investigate chaotic behavior [

A simple road network is used in this study, as shown in Figure

Two-link road network considered in the study.

The evolution of network traffic flow in a two-link network with two intervals is illustrated in Figure

Evolution of network traffic flow on two-link network with a study period divided into two intervals.

The study period is divided into two intervals. The OD demand is fixed. It is assumed that link travel time is related to link traffic volume. The travel time function is expressed as

The evolution process of network flows is shown in Figure

According to the perceived travel cost on day

Assume that (a)

The travel demand in the two intervals can be calculated according to the probabilities of departure choice as follows.

Let

Traffic assignment is carried out in two intervals after determining travel demand. Travelers update their perceived travel costs based on historical and real-time travel information in the chosen departure time. Therefore, the updated perceived travel cost is determined based on the perceived travel cost on day

It is worth noting that for

The traveler’s perceived travel costs for

The utility of link

Route flows in interval 1 can be assigned according to the probability of route choice in interval 1 on day

Let

Then, traffic assignment in interval 2 is given by the following.

Let

Then, the dynamic system model is translated into the following equations.

Note that (_{i} is the Jacobian matrix and_{i} is the eigenvalue of the matrix. In this paper, the maximum Lyapunov exponent is used to judge whether the evolution of network traffic flow is chaotic. If the exponent is greater than 0, the evolution of network traffic flow is considered chaotic.

The traffic network shown in Figure _{0 }is the link free-flow travel time,_{10}) is 22 min and capacity (_{1}) is 1500 veh/h, and for link 2, free-flow travel time (_{20}) is 25 min and capacity (_{2}) is 2000 veh/h. It is assumed that the inherent disutility in interval 1 (_{1}) is 5 and that in interval 2 (_{2}) is 3. The OD demand in the study period is 3000 veh and the study period is 2 hours, which is divided into two intervals. The travelers make route choice in the two-link network and can depart from the origin in any one of the two intervals.

For

The bifurcation diagram formed by traffic-flow evolution of link 1 in interval 1, where

Flow bifurcation diagram with

The results show that the evolutionary processes of traffic demand and traffic flow are similar. The evolution characteristics of traffic demand are shown in Figure

Demand bifurcation diagram with

The different states of traffic-flow evolution with different

System state for different value of

For

The different states of traffic-flow evolution under three types of travel information are shown in Figure

System state for different

To investigate the effect of different types of information on traffic-flow evolution, Figure

The system’s state for different

A phase diagram for each step of the system (i.e., projection of the system attractor on the plane with the two coordinates) is shown in Figure

Projection of chaotic attractor for

In this paper, a day-to-day dynamic evolution model of network traffic flow is formulated considering departure time choice in a two-link network with two-interval analysis period. Traffic-flow evolution under different types of information is investigated using numerical experiments by changing traveler characteristic parameters (

The evolution of network traffic flow as

Overall, the possibility of chaos occurrence is relatively small under the combined historical information and real-time information. However, in the case of chaos occurrence, the complexity of chaotic behavior is relatively small under real-time information alone. This is of great significance for the management and control of network traffic flow using travel information systems. The results show that both historical and real-time information should be used to guide network flows in normal circumstances. When the network traffic flow is unstable, especially when chaos occurs, real-time information should be used to regulate traffic flows.

The evolution of network traffic flow considering traveler’s route and departure time choices is modeled in this paper by introducing a learning mechanism based on travel time information from previous days and that from the same day provided by a real-time informative system. Under the basic framework of this model, other different behavioral assumptions may be adopted, such as bounded rationality. In addition, the proposed model can be used to simulate the conditions where different users are advised by different information systems. For example, one-half of travelers are influenced by historical information and the other half are advised by real-time information.

The findings of this paper are only applicable to a two-link network with two-interval period. The research method of this paper would provide a useful background for analyzing traffic flows on more complex networks. The dynamic evolution characteristics of network traffic flow are analyzed by establishing a larger dimensional nonlinear dynamic model. However, the chaos phenomenon may be more complicated, such as the occurrence of hyperchaos. Some of the preceding topics are currently explored by the authors.

The research presented in this paper is still in the theoretical stage, and therefore its verification using actual data is warranted. If the chaos phenomenon of traffic flow is proved to exist using a large amount of field data, it will raise new research questions, for example, how chaos control can be carried out to make network traffic flow reach a stable state. This definitely would help traffic engineers and practitioners to effectively manage and control road traffic.

The numerical simulations data used to support the findings of this study were supplied by Wensi Chen under license and so cannot be made freely available. Requests for access to these data should be made to Wensi Chen’s e-mail address: 496311795@qq.com.

The authors declare that they have no conflicts of interest.

This research project is financially supported by the National Natural Science Foundation of China (Grant Nos. 51308126, 51378036, and 51308018).