Complex Behaviors of a Simple Traffic Model

Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 952 960 c International Academic Publishers Vol. 46, No. 5, November 15, 2006 Complex Behaviors of a Simple Traffic Model GAO Xing-Ru Department of Physics and Institute of Theoretical Physics, Beijing Normal University, Beijing 100875, China Fundamental Teaching Department, Beijing Union University, Beijing 100101, China (Received December 28, 2005) Abstract In this paper, we propose a modified traffic model in which a single car moves through a sequence of traffic lights controlled by a step function instead of a sine function. In contrast to the previous work [Phys. Rev. E 70 (2004) 016107], we have investigated in detail the dependence of the behavior on four parameters, ω, α, η, and a 1, and given three kinds of bifurcation diagrams, which show three kinds of complex behaviors. We have found that in this model there are chaotic and complex periodic motions, as well as special singularities. We have also analyzed the characteristic of the complex period motion and the essential feature of the singularity. PACS numbers: 89.40.Bb Key words: traffic model, complex behavior, chaos, bifurcation diagram, singularity 1 Introduction In the past few years, the problems of traffic patterns and transportation have been extensively studied, such as traffic jam, the complex behavior of traffic flow, and busroute problems. [1 4] Particularly, the complex behavior displayed in traffic patterns has attracted much attention in statistical mechanics and nonlinear dynamics. Dynamical phase transitions and chaos have been found in traffic flow. [5 8] Various models of traffic systems have been proposed and studied to understand the transportation problems. With the help of these models, some special and complex behaviors of traffic flow in realistic transportation become easy to be understood. Cellular automaton (CA) models play an important role in this field and they have been employed to simulate the traffic interactions between on-ramp and main road. [9 11] CA models are conceptually simple, and easily simulate on computer for numerical investigations. Using the CA models, the conventional phase transitions of traffic flow are investigated. Bus route models exhibit complex behaviors and multiple periodic motions. The bus dynamics of the nonlinear map have been investigated and chaotic motions have been found in the bus route models. [2,3,12,13] Recently a special and simple traffic model has been proposed. [5] In this model, a single car move through a sequence of N traffic lights that turn on and off with the same circular frequency. The n-th traffic light is green if sin(ω n t + φ n ) > 0 and red otherwise. Clearly the circular frequency ω is an important control parameter of the traffic flow. In this sequence of traffic lights, the car can have an acceleration a 1 until its velocity reaches the cruising speed v max. As the car approaches the n-th traffic light with the velocity v, the driver must make a decision whether to step on the brake or not, at the decision point depending on the sign of sin(ω n t+φ n ). If the traffic light is red (sin(ω n t + φ n ) < 0), the driver will step on the brake, the car decreases its speed with a negative acceleration a 2 and finally stop at the n-th traffic light. The distance of the decision point from the traffic light is v 2 /2a 2. If the traffic light is green (sin(ω n t +φ n ) > 0), the car will pass through the traffic light with cruising velocity v max. The velocity v n (n = 1, 2, 3..., N) at the n-th traffic light is governed by the dynamics of nonlinear map. The authors have investigated the dynamical behaviors of the car. They have found the bifurcation diagrams for velocity v n as a function of the circular frequency ω, which implies a double-period bifurcation to chaos with increase of the circular frequency ω. Even though this model is very simple, it displays complex behaviors and chaotic motions. In this paper, we will improve and generalize the above traffic model. As we see, the above traffic model suggests that the time durations of green and red lights are always equivalent in a period. This is not true in the most situations. In fact, on the main road, the duration of green light is generally longer than that of the red light, while on the minor road, the duration of red light is often longer than that of the green light. In our model, we introduce a new parameter α to describe the difference between time durations of green and red traffic lights within a period. Thus the traffic light is characterized by the circular frequency ω and the time ratio α. In our work, we not only investigate the evolution of speed with ω and α, but also the acceleration a 1 and the acceleration ratio η (η = a 1 /a 2 ). We have found a number of interesting results which do not exhibit in the previous models. The paper is organized as follows. In Sec. 2, the traffic model is presented. In Sec. 3, the complex behaviors of the E-mail: gao xingru@yahoo.com.cn

No. 5 Complex Behaviors of a Simple Traffic Model 953 car s motions are shown in which various chaotic motions and complex period behaviors are included. The singularities are investigated in Sec. 4. Finally, the conclusions are given in Sec. 5. 2 Traffic Model We consider the dynamic behaviors of a car moving in a straight street. There are a sequence of traffic lights in the street. The distance between the n-th and (n + 1)-th traffic lights is L n, which is assumed to be independent of n (L n = L). Considering the difference of the duration of green light being on and red one, in our model, we use a function f(t) instead of sin(ω n t + φ n ) to describe the traffic light model, which is shown in Fig. 1, { 1, 0 < t T0, green phase, f(t) = 1, T 0 < t T, red phase, where T 0 is time duration of green light in a period T. Therefore, (T T 0 ) is the duration of the red light. We define the time ratio α, which reflects the configuration of the intersection, α = T 0 T. Obviously, α takes value: 0 α 1. If the green phase is equal to the red phase (T 0 = T/2), the time ratio α = 0.5. If the time ratio α is equal to 1 (T 0 = T ), it means that the green light is always on and the driver will never step on the brake, thus the car will pass every traffic light with the cruising speed v max as if there is no traffic light. When a main road intersects a minor road, α > 0.5 (T 0 > T/2 namely the green phase is more than red phase) is assigned to the main road. frequency ω and the other one is the time ratio α. In the previous model, [5] the function f(t) = sin(ωt + φ) is used as the traffic light control function. It means that the time ratio α is equal to 0.5. Hence, the previous model is a special case of our new model at α = 0.5. The initial velocity of the car is υ 0 and the car accelerates with a 1 until the velocity reaches the cruising speed υ max. Then the car moves at a constant speed υ max. When the driver steps on the brake, the car has a negative acceleration a 2 until it stops. We define the acceleration ratio η = a 1 /a 2 and η is assigned a value between 0 1. As the car approaches the traffic light with the velocity υ the driver must make a decision whether to step on the brake or not at the distance x = υ2 2a 2, depending on the sign of function f(t). There are three possible situations at the decision point: the first is that the driver continues to pass the traffic light with the cruising speed υ max if the function f(t) takes +1, which means the traffic light is green. The second is that the driver steps on the brake to stop the car completely until the light turns green again. The third is that the driver brakes the car and accelerates again as the light turns green before the car stops completely. The three types of trajectories of the car are shown in Fig. 2. [5] Fig. 2 The three possible situations at the decision point. (i) Continuing; (ii) Braking to stop; (iii) Braking and accelerating again as the light turns green before stopping completely. Fig. 1 The traffic light model. (i) 0 < t T 0, f(t) = 1, the traffic light is green; (ii) T 0 < t T, f(t) = 1, the traffic light is red. The relation between the circular frequency ω and the period T is shown as follows: ω = 2π T. Therefore, in our model, there are two parameters to express the properties of traffic light. One is the circular The initial velocity υ 0 is not important, so in our investigation, we do not pay much attention to υ 0 and suppose υ 0 = 0. In our traffic model, there are two parameters to describe the motions of the car, viz. acceleration a 1 and the acceleration ratio η. With a similar derivation to Ref. [5], it is easy to construct the iterated map that relates successive crossing of the traffic lights. The velocity is v n at the n-th traffic light. After crossing the n-th traffic light, the driver accelerates the car with a 1, the car reaches v max, then moves

954 GAO Xing-Ru Vol. 46 at cruising speed v max until the decision point. At the decision point, the driver makes a decision continuing or braking according to the (n + 1)-th traffic light. If the function f(t d ) = 1 (t d is the time of the car reaching the decision point), the traffic light is green, the car continues to cross the traffic light with v max, v n+1 = v max. If the function f(t d ) = 1, the traffic light is red, the driver steps on the brake. The speed decreases with a 2. There are two possible situations depending on the color of traffic light. Fig. 3 An iterated map for the speed v n at the n-th traffic light for v 0 = 0, ω = 5.6, α = 0.7, a 1 = 1.1, and η = 0.165. In one case, the traffic light is always red until the car stops completely. So the car s velocity at the (n + 1)-th traffic light is zero, v n+1 = 0. In the other case, the traffic light turns green before the car stops completely, the car starts to accelerate with a 1 before it reaches the traffic light. Hence, we must calculate an extra time t = v max /a 2 at which the car will take to reach the traffic light and stop. At the same time, we calculate the time t g of the traffic light turns green again, t g = [ In t ( td ) + 1 ] T T Then we compare t d + t with t g. If t g > t d + t, the traffic light is always red until the car stops. Therefore the car s velocity at the (n + 1)-th traffic light is zero, v n+1 = 0. Otherwise, t g < t d + t, the traffic light turns green before the car stops completely. The car starts to accelerate with a 1 again. There are also two kinds of cases as follows. One case is that the car reaches v max before the traffic light. The car continues to cross the traffic light with v max, v n+1 = v max. The other case is that the car s velocity does not reach v max at the traffic light. The car will cross the traffic light with v n+1 = vg 2 2a 1 (L n x g ), where x g and v g are the position and velocity of the car when the traffic light turns green again. [5] Depending on the above iterated map, we can get the conclusion that there are three possible situations of the velocity v n+1 at the (n + 1)-th traffic light. (i) v n+1 = v max ; (ii) v n+1 = 0; (iii) v n+1 is the value between 0 and v max. Figure 3 is an iterated map for the speed v n at the n-th traffic light for a certain parameters (v 0 = 0, ω = 5.6, α = 0.7, a 1 = 1.1, and η = 0.165). In our model, if we choose the value of α to be 0.5, we can get the same results as the previous work. At the same time, we also get some other different results when we choose α = 0.8 instead of α = 0.5. In the previous work, [5] the author only studied the chaos motions of the car with the circular frequency ω of the traffic lights. In actual traffic control strategies, the circular frequency ω of the traffic lights is not the only parameter. The time ratio α of the traffic lights is another very important parameter. On the other hand, the car s acceleration a 1 and the acceleration ratio η also are very important variables in traffic control strategies. We believe from the above iterated map that the following four parameters ω, α, η, and a 1 may strongly affect the motion behaviors of the car. 3 Period and Chaos Behaviors For convenience, we assume that the maximum velocity of the car is a given value and define the cruising time as T c = L/V 2 max. We can normalize v = V/V max, a 1 = A 1 L/V 2 max, a 2 = A 2 L/V 2 max, and ω = ΩT c (where V, A 1, A 2, and Ω are actual variables), therefore all variables and parameters are dimensionless. [5] So the maximum velocity of the car is equal to 1 (v max = 1). In our research, there must be enough traffic lights N in the straight street, we choose N = 200. As usual, the transient behavior has been ignored. Now we study how these four parameters (ω, α, η, a 1 ) determine the dynamic behaviors of the car. In the four parameters, we choose three of them to be fixed values and only one parameter to vary in a certain range. We find chaotic behaviors in three cases and one complex period behaviors for a certain ranges of the time ratio α. We also find some special singularities and study their characteristics. 3.1 Chaotic Behaviors Now, we find the chaotic behaviors in the following three cases. The first case is that the circular frequency

No. 5 Complex Behaviors of a Simple Traffic Model 955 of traffic light ω varies in a range, and the other three parameters (a 1, η, α) are respectively the fixed values. The second case is that the acceleration a 1 varies in a range, and the other three parameters (ω, η, α) are respectively the fixed values. The third case is that the acceleration ratio η varies in a range, and the other three parameters (a 1, ω, α) are respectively the fixed values. It means that we can find the chaotic motions when any one of the three parameters (ω, a 1, η) is varying in a range and the other parameters are the fixed values. Firstly, when the circular frequency of traffic lights ω is varying from 0 to 6.3 and the other three parameters (a 1, η, α) take the following values: the acceleration a 1 = 1.1, the acceleration ratio η = 0.165, the time ratio of traffic light α = 0.8 (It means that the time of green light is longer than the red light.), we find three chaos as shown in the bifurcation diagrams of Figs. 4(a), 4(b), and 4(c). As we see, the chaotic regions appear after a period-doubling bifurcation as frequency ω decreases. Figure 4(d) is the zoom for Fig. 4(c). It displays an intrinsic and similar chaotic structure. Fig. 4 The bifurcation diagram for (a) ω = 2.9 3.15; (b) ω = 4.62 4.67; (c) ω = 5.4 6.3; (d) Zoom for (c). The other parameters are a 1 = 1.1, η = 0.165, and α = 0.8. Besides the above chaotic regions, we can also find the other chaotic regions as the circular frequency ω increases. As we see in Fig. 4, the car can have different velocities for a given ω. In order to use an easy way to analyze the car s complex behaviors, we define an integer number m to describe the number of allowed values of velocities the car can reach at the n-th traffic light for a given circular frequency. For example, for a given ω = 3.1, the car can have a unique velocity, for a given ω = 3.05, the car can have two allowed values of velocities, so m equals 1 and 2 respectively. In our iterated process, the car moves through a sequence of at least N = 200 traffic lights. We remove the front 150 traffic lights transients. So the maximum value of the number m is 50 (m max = 50). The value of m becomes greater and greater from the period-doubling bifurcation to the chaotic region. Figure 5 displays the number m varying with the circular frequency ω. We can also easily obtain the car s complex behaviors in the lower circular frequency ω = 0 30. If the circular frequency ω

956 GAO Xing-Ru Vol. 46 increases enough, the car s complex behaviors disappear. Fig. 5 The number m varies with the circular frequency ω (0 30). The other parameters are a 1 = 1.1, η = 0.165, and α = 0.8. Secondly, we consider the relation between v n and a 1. Let us choose, for example, the value of ω to be 5.6, the acceleration ratio η = 0.165 and the time ratio α = 0.8 and investigate the car s velocity varies with the car s acceleration a 1. Figure 6(a) is one of bifurcation diagrams in which we also find chaos region. From Fig. 6(b), it is easy to see that most of chaotic region of the car s acceleration is in 0 3 if the circular frequency ω is not too large. Thirdly, we let the circular frequency ω be 5.6, the time ratio of traffic light α equals 0.8, and the car s acceleration a 1 is one of the values in the chaotic region, such as a 1 = 2.1. We consider the values of the acceleration ratio η vary from 0 to 1. The bifurcation diagram is shown in Fig. 7(a). If we change one of the three parameters ω, α, and a 1, we can also obtain the similar bifurcation diagrams. However it is difficult to find the chaotic motions if the acceleration ratio η is more than 0.5. Figure 7(a) shows the evolutive process of the period-doubling bifurcation to the chaos when the acceleration ratio η changes from 0.25 to 0.4. Fig. 6 (a) The bifurcation diagram for a 1 = 0.3 2.0 and the other parameters are ω = 5.6, η = 0.165, and α = 0.8. (b) The number m varies with the acceleration a 1(0 5.0). The other parameters are ω = 5.6, η = 0.165, and α = 0.8.

No. 5 Complex Behaviors of a Simple Traffic Model 957 Fig. 7 The evolutive process of bifurcation diagram with a 1 when η varies within a certain range (η = 0.2 0.5) and ω = 5.6, α = 0.8. (a) a 1 = 2.1; (b) a 1 = 2.55; (c) a 1 = 2.35; (d) a 1 = 2.25. In addition, in Fig. 7 we also give the bifurcation diagrams in different values of a 1. Figure 7(b) indicates that there is no chaotic motion at a 1 = 2.55. As the value of a 1 decreases gradually, the two bifurcation branches of bifurcation are close to each other and become a mixture at last. When the value of a 1 decreases to 2.1, we get the bifurcation diagram Fig. 7(a). In sum, the bifurcation diagrams for the speed can be found when one of the three parameters (the circular frequency ω, the car s acceleration a 1, and the acceleration ratio η) is changed in a certain range. However, if the time ratio of traffic light α is changed from 0 to 1 and the other three parameters are respectively the fixed values, we have not found any bifurcation diagram. The details will be discussed in the next section. 3.2 Complex Period Behaviors Fig. 8 (a) The relation between the time ratio of traffic light α and the car s behavior, the other parameters are ω = 5.6, η = 0.165, and a 1 = 1.1; (b) is the detail of (a). Now, let us study the relation between the time ratio of traffic light α and the car s behavior. Suppose the values of ω, a 1, and η are respectively 5.6, 1.1, and 0.165, figure 8(a) displays the relation of the car s behaviors with the change of α. Figure 8(b) is the detail of Fig. 8(a). It shows a special kind of complex behavior which is different from chaos. The car s velocity is always zero when the car arrives at any traffic light as the value of α is less than 0.229. When the value of α is located between 0.229 and 0.881, the iterated map does not change with α, hence the car s behavior is not related with the change of α. When the value of α is located between 0.881 and 0.971, the car s behavior is very sensitive to the change of α. The car will pass through all traffic lights as the value of α is more than 0.971. On the other hand, we find some special and regular diagrams when we change the values of parameters. The complex period behaviors emerge. Figure 9(a) shows one of them. The parameters are respectively ω = 8.105, η = 0.5,

958 GAO Xing-Ru Vol. 46 a 1 = 1.1 and the range of α is from 0.31 to 0.44 in Fig. 9(a). This is also a bifurcation diagram, but it is not period-doubling bifurcation diagram as usual. There are some special characteristics in Fig. 9(a). Fig. 9 A special and regular diagram for ω = 8.105, η = 0.5, a 1 = 1.1 and three linear relations. (a) A special bifurcation diagram; (b) The linear relation between m and α; (c) The linear relation between v max and α; (d) The linear relation between v min and α. Now we investigate the properties of Fig. 9(a). Obviously, it is symmetric about v n = 0.38, and the number m of allowed velocities, which is shown by horizontal lines, corresponding to a given α. Three linear relations are found in Fig. 9(a). The first linear relationship is that the horizontal lines number m increases linearly with the time ratio of traffic light α when the value of α is located between the range of 0.335 0.413 as shown in Fig. 9(b). The linear relation is fitted as follows: m = 88.75 + 312.84α. Figures 9(c) and 9(d) show the two other linear relations. In the symmetric diagram of Fig. 9(a), the minimum velocity v min decreases linearly and the maximum velocity v max increases linearly with the increase of α between the range of 0.318 0.426. Both of linear relations are fitted as follows: v max = 0.26 + 2.09α, v min = 0.83 1.54α. Fig. 10 The iterated map for ω = 8.105, η = 0.5, a 1 = 1.1, and α = 0.35. Figure 10 shows a typical iterated map for the car s speed v n at the n-th traffic light when the four parameters are respectively ω = 8.105, η = 0.5, a 1 = 1.1, α = 0.35. It represents a special period behavior. Until now, we have not found any chaotic motion when the value of α is

No. 5 Complex Behaviors of a Simple Traffic Model 959 changed from 0 to 1 and the other three parameters (ω, a 1, η) are chosen some fixed values. case at ω c. 4 Singularity We also find some singularities when the four parameters are respectively special values. In Fig. 11, such a singularity diagram appears at a 1 = 1.1, η = 0.165, α = 0.5 (The latter means that the duration of green light being on is equal to the red.) and ω = 9.1 9.2. It is a new type of bifurcation diagram. The number m of allowed velocities changes gradually with ω, as shown in Fig. 12(a). The number m reaches the maximum at ω c = 9.162 083 620 352 2. From now on, we study the characteristic of the system around the singularity ω c. Fig. 11 A singularity for ω = 9.1 9.2 and the other parameters are respectively a 1 = 1.1, η = 0.165, and α = 0.5. We define the increment ω m = ω(m+2) ω(m). Figure 12(b) indicates that there are three different linear regions. The linear ranges of the circular frequency ω are (i) ω = 9.122 115 9.162 071; (ii) ω = 9.162 071 9.174 899; (iii) ω = 9.174 899 9.195 891. We define δ m as same as M.J. Feigenbaum in Logistic map: δ m = ω m ω m 2 ω m+2 ω m = ω m 2 ω m, m = 3, 5, 7,... We calculate the values of δ m when ω approaches ω c from left and right respectively. In the first case, the value of m increases with ω, and the value of δ m approaches 1.741 372 375. In the second case, the value of m decreases with ω, the value of 1/δ m approaches 1.764 972 660. From the above analyses, we can guess that the limit of δ m in the first case is equal to the limit of 1/δ m in the second Fig. 12 A singularity properties for (a) the relation of the number m and ω; (b) three different linear regions of the number m and ω m. Figure 13 shows the relation between the maximum values v max of the number m of allowed velocities and ω. It is very easy to determine the value of ω c in Fig. 13. Now we calculate the fixed velocities at ω c = 9.162 083 620 352 2. In the same way, we define the increment v m = v(m + 2) v(m) and ε m as follows: ε m = v m v m 2 v m+2 v m = v m 2 v m, m = 3, 5, 7,... We find that there is a special value v c = 0.724 165 435 892 152. When the velocity increases and approaches v c, the value of ε m is equal to 1.751 518 267. At the same time, when the velocity decreases and approaches v c, the value of 1/ε m is equal to 1.751 527 318. We can also guess that the limits of ε m and 1/ε m are equal at v c. From the above discussion, it seems probable that at singularity (ω c = 9.162 083 620 352 2, v c = 0.724 165 435 892 152), the limit δ m may be expected to be equal to the limit of ε m.

960 GAO Xing-Ru Vol. 46 have found the behaviors of motion are strongly dependent on the parameters ω, α, η, and a 1. We clearly understand that realistic situations are not as simple as our model. In our model, there is only one car in a straight street, the distance between traffic lights is constant and all of traffic lights have same circular frequency and time ratio. Actually, the traffic situation is more complex. If we extrapolates our model to actual traffic situation, other parameters are necessary, such as the distance of traffic lights, the types of car, the personality of driver, etc. With our simple model, we already have observed some essential features. Fig. 13 The relation between the maximum values v max and ω near the singularity ω c. 5 Conclusion and Discussion Although our traffic model is very simple, we have found very complex behaviors in such a simple system, in which there are chaos and complex period motions. We Acknowledgments Prof. Z.R. Yang provided helpful guidance to this work. We are very thankful to Prof. Z.R. Yang and grateful to Profs. Z.G. Zheng, Z. Gao, and W.A. Guo, who provided many good suggestions to this work. We also acknowledge fruitful discussions with Drs. J.X. Le, X.M. Kong, X.H. Li, and J.Q. Tao. References [1] T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331. [2] T. Nagatani, Phys. Rev. E 66 (2002) 046103. [3] D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. [4] D. Chowdhury, L. Santen, and A. Schadscheider, Phys. Rep. 329 (2000) 199. [5] B.A. Toledo,V. Munoz, J. Rogan, and C. Tenreiro, Phys. Rev. E 70 (2004) 016107. [6] T. Nagatani, Phys. Rev. E 68 (2003) 036107. [7] T. Nagatani, Phys. Rev. E 63 (2001) 036116. [8] H.J.C. Huijberts, Physica A 308 (2002) 489. [9] Rui Jiang, Qing-Song Wu, and Bing-Hong Wang, Phys. Rev. E 66 (2002) 036104. [10] Ding-Wei Huang and Wei-Neng Huang, Phys. Rev. E 67 (2003) 068101. [11] Rui Jiang, Qing-Song Wu, and Bing-Hong Wang, Phys. Rev. E 67 (2003) 068102. [12] T. Nagatani, Physica A 319 (2002) 568. [13] T. Nagatani, Physica A 323 (2003) 686.