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Fluctuation of tour time induced by interactions between cyclic trams
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Physica A 331 (2004) 279 – 290
www.elsevier.com/locate/physa
Fluctuation of tour time induced by interactions between cyclic trams Takashi Nagatani∗ Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan
Abstract We study the dynamical behavior of N trams which move around the tram stops on a cyclic route repeatedly. We present the dynamical model for the cyclic trams. When a tram catches another tram, the tram restarts after delay time Tmin and keeps the minimal time headway Tmin . The distinct dynamical states (the regular, periodic, and chaotic motions) are found by varying loading parameter , delay time Tmin , and number M of tram stops. It is shown that the dynamical transitions occur from the regular motion, through multiply periodic motions, and to the chaotic motion. In the periodic and chaotic motions, the tour and arrival times of trams +uctuate highly. It is shown that the tram-stop’s number M has an important e.ect on the tour time of trams. The phase diagram (region map) is found. c 2003 Elsevier B.V. All rights reserved. PACS: 89.40.+k; 05.70.Fh; 05.90.+m Keywords: Dynamical transition; Chaos; Tram; Phase diagram
1. Introduction Recently, the transportation problems have attracted much attention [1–9]. The vehicular tra;c +ow has been investigated by various models [1–6]. There have been some corresponding studies of buses [10–17]. The bus-route system is a typical dynamical system of interacting buses and passengers. The dynamics of a bus route is di.erent from the tra;c dynamics since buses interact with passengers at designated bus stops in addition to the interaction between buses. ∗
Fax: +81-53-478-1048. E-mail address: [email protected] (T. Nagatani).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.07.007
280
T. Nagatani / Physica A 331 (2004) 279 – 290
In the system with many buses, the dynamical phase transition to the inhomogeneous jammed (bunching) phase has been found [10–17]. If a bus is delayed by some +uctuation, the time headway (gap) between itself and its predecessor becomes larger than the initial time headway since it has to pick up more passengers. The slowly moving delayed bus slows down the buses behind it. As a result, all buses behind it bunch together. The single cyclic bus system also exhibits the nontrivial delay transition [18,19]. In a single cyclic bus system with many bus stops, the bus interacts with prospective passengers waiting at bus stops. This interaction induces the dynamical phase transition between the delay and scheduled time phases. The delay transition is fundamentally di.erent from the bunching transition of many buses. The single cyclic bus model has been extended to the shuttle-bus system with a few buses [20,21]. The buses shuttle between a starting point and a destination on a route. There are the two interactions: the one is the interaction between buses and the other is the interaction between a shuttle bus and passengers waiting at the starting point. It has been found that the two interactions induce a complex behavior of buses. The shuttle-bus system exhibits the dynamical transitions to the chaotic and periodic motions. Also, the motions of trams (streetcars) are similar to those of shuttle buses. The trams move around the stops repeatedly. The passengers waiting at a tram stop board the coming tram. Generally, the perspective passengers waiting at a tram stop increases accordingly as the time headway between a tram and the next tram increases. The tram cannot pass over the other tram. It will be expected that the motions of trams exhibit a complex behavior by the interaction between trams and passengers. The complex motions of trams will induce the large +uctuation of arrival time and riding passengers. It is very important to estimate the arrival time and time headway of trams for public demand. In this paper, we study the dynamical behavior of a few trams on a cyclic route. We present the dynamical model for the cyclic trams. We investigate the +uctuation of arrival time and tour time. We clarify that the trams exhibit the periodic and chaotic motions. We study the dynamical phase transitions among the distinct dynamical states. We present the phase diagram (region map) of the distinct dynamical states. We Dnd the optimal time headway for keeping the well-controlled operation. 2. Model We present the dynamical model of N trams moving on a cyclic route with M tram stops. Trams start at tram stop m = 1, move around the route, stop at all the tram stops m = 1; : : : ; M , and return to tram stop m = 1. All the trams move around the cyclic route repeatedly without passing other trams. The model is deDned on a Dnite one-dimensional lattice with the periodic boundary. Each lattice site is labeled by number m running from 1 to M . A site represents a tram stop. All the trams are numbered from 1 to N in regular order. The distance between a stop m − 1 and its next stop m is deDned by Lm .
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If a tram catches an other tram, its tram stops and restarts after delay time Tmin . The tram does not change the speed except for stopping at the tram stops. The tram keeps constant average speed until it stops. The new passengers arrive at all the tram stops at the rate . So Gt is the number of passengers that have arrived since its predecessor has left the tram stop, where Gt is the time headway between itself and its predecessor. We assume that new passengers try to board the tram after the passengers left the tram. The time ts; m is the amount of time needed to leave the passengers at the tram stop m. The parameter is the time it takes one passenger to board the tram, so Gt is the amount of time needed to board all the passengers at the tram stop. The arrival time t1 (m; n + 1) of tram 1 on tram stop m at trip n + 1 is given by t1 (m; n + 1) = t1 (m; n) +
M
m−1
m=1
k=1
L + ts; m + [t1 (k; n + 1) − tN (k; n)] V
M [t1 (k; n) − tN (k; n − 1)] + 1 (m; n) ; +
(1)
k=m
M where L = m=1 Lm is the length of the route and V is the average speed of trams. 1 (m; n) is a noise of tram 1 at on tram stop m at trip n. The noise is due to a +uctuation of passengers waiting on tram stops. The mean and correlation of the noise are given by 1 (m; n) = 0
and
1 (m; n)i (m ; n ) = a2 1i mm nn :
If tram 1 catches tram N (t1 (m; n + 1) ¡ tN (m; n)) at tram stop m, tram 1 restarts after delay time Tmin : t1 (m; n + 1) = tN (m; n) + Tmin :
(2)
Similarly, the arrival time ti (m; n + 1) of tram i(i ¿ 2) on tram stop m at trip n + 1 is given by ti (m; n + 1) = ti (m; n) +
+
M
M
m−1
m=1
k=1
L ts; m + [ti (k; n + 1) − ti−1 (k; n + 1)] + V
[ti (k; n) − ti−1 (k; n)] + i (m; n) :
(3)
k=m
If tram i catches tram i − 1(ti (m; n + 1) ¡ ti−1 (m; n + 1)), tram i restarts after delay time Tmin : ti (m; n + 1) = ti−1 (m; n + 1) + Tmin :
(4) M
We assume that the amount of time needed to leave the tram is constant m=1 ts; m = M constant. By dividing time by the characteristic time L=V + m=1 ts; m , one obtains the
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T. Nagatani / Physica A 331 (2004) 279 – 290
governing equations of dimensionless arrival times m−1
T1 (m; n + 1) = T1 (m; n) + 1 +
[T1 (k; n + 1) − TN (k; n)]
k=1
+
M
[T1 (k; n) − TN (k; n − 1)] + 1 (m; n) ;
(5)
k=m
if T1 (m; n + 1) ¡ TN (m; n) ;
T1 (m; n + 1) = TN (m; n) + Tmin Ti (m; n + 1) = Ti (m; n) + 1 +
m−1
(6)
[Ti (k; n + 1) − Ti−1 (k; n + 1)]
k=1
+
M
[Ti (k; n) − Ti−1 (k; n)] + i (m; n) ;
(7)
k=m
Ti (m; n + 1) = Ti−1 (m; n + 1) + Tmin where
Ti (n) ≡ ti (n)
L=V +
L=V +
Tmin ≡ tmin
M
ts; m
a =a
L=V +
L=V + M m=1
; M
ts; m
;
m=1
ts; m
;
ts; m
m=1 M
= i (m; n)
(8)
m=1
i (m; n)
if Ti (m; n + 1) ¡ Ti−1 (m; n + 1) ;
;
2
i (m; n)j (m ; n ) = a ij mm nn : The tram’s motion depends on parameters , Tmin , number M of tram stops, and tram’s number N . With increasing loading parameter , the tram slows down since more passengers board the tram. With decreasing delay time Tmin , the following tram becomes faster since less passengers board the tram. Thus, a tram interacts other trams. It is expected that the trams exhibit a complex behavior and dynamical transitions. 3. Simulation result We study the tram’s behavior by the use of iterates of Eqs. (5)–(8). First, we study the dynamical behavior of two trams without noises. The initial condition is given by T1 (0) = 0;
T2 (0) = 0:5;
T1 (1) = 1:0 and T2 (1) = 1:5 :
(9)
Fig. 1 shows the time evolutions of the tour time (one period) GT 1(n) and the time headway GT 12(n) for two values of delay time Tmin under the constant value of
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Fig. 1. Time evolutions of the tour time (one period) GT 1(n) and the time headway GT 12(n) for two values of delay time Tmin under the constant value of loading parameter where GT 1(n) ≡ T1 (n) − T1 (n) − T1 (n − 1) and GT 12(n) ≡ T1 (n). Diagrams (a) and (b) are obtained for Tmin = 0:4 and 1.0 where = 0:15, tram’s number N = 2, and stop’s number M = 4.
loading parameter where tram stop’s number M = 4, GT 1(n) ≡ T1 (n) − T1 (n − 1), and GT 12(n) ≡ T2 (n) − T1 (n). Diagrams (a) and (b) are obtained for Tmin = 0:4 and 1.0 where = 0:15. For (a) Tmin = 0:4, two trams move with a constant value of tour time and tram 2 always follows tram 1 with constant value Tmin of time headway. For (b) Tmin = 1:0, the motions of two trams exhibit an irregular oscillation. The oscillation may exhibit a chaotic behavior. The tour time and time headway oscillate largely and irregularly. We calculate the variation of tour time GT 1(n) by increasing return n to study the dynamical transitions to the complex motions. Fig. 2 shows the plot of the values of tour time GT 1(n) against delay time Tmin from su;ciently large return n = 100 to 300 where tram’s number N = 2 and = 0:15. Diagrams (a) – (c) are obtained for tram-stop’s number M = 2– 4. In the case of (a) M = 2, when delay time Tmin is less than the critical value Tmin; c = 0:247, the tour time converges to the constant value. Tram 2 follows tram 1 with the constant time headway Tmin . If delay time Tmin is higher than the critical value Tmin; c = 0:247, the tour time of trams oscillate periodically. The dynamical transition from the regular motion to the periodic motion occurs at Tmin; c =0:247. The tour time exhibits the complex behavior with increasing the delay time Tmin . For small values of tour time, the tour time decreases accordingly as the delay time increases. However, the tour time increases with the delay time when the delay time is higher than about one. In the case of (b) M = 3, when delay time Tmin is less than the critical value Tmin; c = 0:361, the tour time converges to the constant value. The dynamical transition from the regular motion to the periodic motion occurs at Tmin; c = 0:361. The transition point is higher than that in Fig. 2(a) of M = 2. The +uctuation of the tour time is larger than that in Fig. 2(a). The dynamical behavior is qualitatively similar to that in Fig. 2(a) but is quantitatively di.erent from that in Fig. 2(a). In the case of (c) M = 4, when delay time Tmin is less than the critical value Tmin; c = 0:484, the tour time converges to the constant value. The dynamical transition
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T. Nagatani / Physica A 331 (2004) 279 – 290
Fig. 2. Plot of the values of tour time GT 1(n) against delay time Tmin from su;ciently large return n = 100 to 300 where = 0:15 and tram’s number N = 2. (a) Tram-stop’s number M = 2. (b) Tram-stop’s number M = 3. (c) Tram-stop’s number M = 4.
from the regular motion to the periodic motion occurs at Tmin; c = 0:484. The transition point is higher than those in Fig. 2(a) and (b) of M = 2 and 3. The +uctuation of the tour time is larger than those in Fig. 2(a) and (b). The dynamical behavior is qualitatively similar to those in Fig. 2(a) and (b) but is quantitatively di.erent from those in Fig. 2(a) and (b). We calculate the Liapunov exponent to study the chaotic motion. Fig. 3(a) – (c) shows the plot of Liapunov exponent against delay time Tmin for the same values of parameters in Fig. 2(a) – (c). In Fig. 3(a) of tram-stop’s number M = 2, the value of Liapunov exponent changes from a negative value to a positive value at Tmin; c =0:585 and from a positive value to a negative value at Tmin = 0:813. Therefore, the irregular motion in the region between 0:585 ¡ Tmin ¡ 0:813 exhibits the chaos. In Fig. 3(b) of tram-stop’s number M = 3, the value of Liapunov exponent changes from a negative value to a positive value at Tmin; c = 0:642 and from a positive value to a negative value at Tmin = 0:996. Therefore, the irregular motion in the region between 0:642 ¡ Tmin ¡ 0:996 exhibits the chaos. In Fig. 3(c) of tram-stop’s number M = 4, the value of Liapunov exponent changes from a negative value to a positive value at Tmin; c = 0:708 and from a positive value to a negative value at Tmin = 1:233. Therefore, the irregular motion in the region between 0:708 ¡ Tmin ¡ 1:233 exhibits the chaos.
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Fig. 3. Plot of Liapunov exponent against delay time Tmin where = 0:15 and tram’s number N = 2. Plots (a) – (c) correspond, respectively, to those in Fig. 2(a) – (c). (a) Tram-stop’s number M = 2. (b) Tram-stop’s number M = 3. (c) Tram-stop’s number M = 4.
Fig. 4. (a) Enlargement of Fig. 2(b) for 0:35 ¡ Tmin ¡ 0:60. (b) Enlargement of Fig. 2(c) for 0:45 ¡ Tmin ¡ 0:70.
The transition point to chaos increases with the tram-stop’s number. Also, the region of the chaotic motion increases with the tram-stop’s number. Fig. 4(a) shows the enlargement of Fig. 2(b) between 0:35 ¡ Tmin ¡ 0:60. Many transitions occur among periodic motions of di.erent periods. Fig. 4(b) shows the enlargement of Fig. 2(c) between 0:45 ¡ Tmin ¡ 0:70. The Dne structure of periodic motions in the tour times is deDnitely di.erent from that in Fig. 4(a).
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T. Nagatani / Physica A 331 (2004) 279 – 290
Fig. 5. Plots of average and root-mean-square (rms) of tour time against delay time Tmin for tram-stop’s number M = 4 where = 0:15 and tram’s number N = 2. Point 1 indicates the transition point between the regular and periodic motions. Point 2 indicates the transition point between the periodic and chaotic motions. The chaotic motion appears in the region between points 2 and 3. Point 4 indicates the minimum value of the tour time.
Fig. 6. Plot of the values of tour time GT 1(n) against loading parameter from su;ciently large return n = 100 to 300 under the constant value Tmin = 0:5 of delay time for stop’s number M = 6.
Fig. 5 shows the plots of the average value and root-mean-square (rms) of tour time against delay time Tmin for Fig. 2(c). The mean tour time decreases with increasing the delay time until the transition point 1 at which the regular motion changes the periodic motion. In the periodic motion, the mean tour time increases slightly with the delay time. Then, the average value decreases discontinuously at transition point 2 where the dynamical transition occurs to the chaotic motion. Also, there is the gap in the tour time at the transition point 3 where the chaotic motion changes the periodic motion again. The tour time takes the minimal value of the average at the point 4 where the trams move chaotically. The rms increases linearly with the delay time until point 5. At point 5 where the motion of trams is in the chaotic state, the rms increases abruptly and there is a gap in the rms. Thus, the mean value exhibits the complex behavior due to the periodic and chaotic motions. Fig. 6 shows the plot of the values of tour time GT 1(n) against loading parameter from su;ciently large return n = 100 to 300 under the constant value Tmin = 0:5 of
T. Nagatani / Physica A 331 (2004) 279 – 290
287
Fig. 7. Phase diagram (region map) in phase space (Tmin ; M ) for two trams and loading parameter = 0:15 where Tmin is the delay time and M is the number of tram stops. The regular motion in which the tour time converges to a constant value is indicated by region 1. The periodic motion appears in region 2. The chaotic motion appears in region 3.
Fig. 8. E.ect of noises on the tour time. Plot of tour time GT 1(n) against delay time Tmin from su;ciently large return n = 100 to 300 where = 0:15, tram’s number N = 2, and stop’s number M = 4 at noise’s amplitude a = 0:05. Compare this with Fig. 2(c) of no noise.
delay time for stop’s number M = 6. The tour time increases with loading parameter according as the multiple transitions occur among various periods. We study the phase diagram (region map) for the distinct dynamical states and the dynamical transitions. Fig. 7 shows the phase diagram (region map) in phase space (Tmin ; M ) for two trams and loading parameter = 0:15 where Tmin is the delay time and M is the number of tram stops. The regular motion in which the tour time converges to a constant value is indicated by region 1. The periodic motion appears in region 2. The chaotic motion appears in region 3. We Dnd that the complex behavior of trams depends highly on delay time Tmin , tram-stop’s number M , and loading parameter . We study the e.ect of noises on the motions of trams. Fig. 8 shows the plot of tour time GT 1(n) against delay time Tmin from su;ciently large return n=100 to 300 where =0:15, tram’s number N =2, and stop’s number M =4 at noise’s amplitude a =0:05.
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T. Nagatani / Physica A 331 (2004) 279 – 290
Compare this with Fig. 2(c) of no noise. The trams exhibit the stochastic motions over a wide range by adding the noises to the deterministic model. The periodic motions disappear. However, the average value of tour time changes little. In order to operate the trams in the optimal state, it is necessary to be in the regular state just below the transition point between the regular and periodic motions. In the regular state just below the transition point, the trams move with the minimal tour time within the region of the regular motion. 4. Analysis We present the theoretical analysis to determine the transition point from the regular motion to the periodic motion. We consider such condition that two trams keep the regular motion. In the regular motion, tram 2 follows tram 1 with keeping constant time headway Tmin . Then, the tour time GT 1 of tram 1 is obtained from Eq. (5) 1 − MTmin : (10) GT 1 = 1 − M The tour time of tram 2 is given by GT 2 = 1 + (1 + M)Tmin :
(11)
If tour time GT 1 of tram 1 is higher than GT 2 of tram 2, tram 2 always catches tram 1 and tram 2 follows tram 1 with keeping time headway Tmin . Thus, one obtains the following condition to maintain the regular motion: M : (12) Tmin ¡ 1 + M − (M)2 The transition point Tmin; c from the regular motion to the periodic motion is given by M : (13) Tmin; c = 1 + M − (M)2 Fig. 9 shows the plot of transition point against tram-stop’s number M for = 0:09; 0:12, and 0.15. The transition point increases accordingly as stop’s number M increases under the condition of constant loading parameter. The solid curve represents
Fig. 9. Plot of transition point between the regular and periodic motions against tram-stop’s number M for loading parameter = 0:09; 0:12, and 0.15 where N = 2. The circles indicate the simulation result. The solid lines represent the theoretical result.
T. Nagatani / Physica A 331 (2004) 279 – 290
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the theoretical transition point (13). The circular point indicates the simulation result. The theoretical result agrees with simulation result. When M = 1, Eq. (13) agrees with that obtained for two shuttle buses [20]. In Eq. (10), the tour time diverges when loading parameter approaches to 1=M . Also, under the condition of constant loading parameter, the tour time diverges when stop’s number M is higher than 1=. Thus, one obtains the following relation for Dnite tour time: 1 M¡ : (14) We showed that the tour time diverges when stop’s number M is equal to or higher than 7 for = 0:15 in the simulation of Section 3. This simulation result is consistent with Eq. (14). 5. Summary We have studied the dynamical behavior of trams moving around the tram stops repeatedly. We have presented the dynamical model of N trams, which is described by four parameters: loading parameter , minimum headway Tmin , tram-stop’s number M , and tram’s number N . We have found that there are the distinct dynamical states: the regular motion, the periodic motion, and the chaotic motion. We have shown that the dynamical transitions occur among the distinct dynamical states with varying their parameters. Especially, we have shown that the number of tram stops has the important e.ect on the motion of trams. We have presented the theoretical result to predict the transition point between the regular and periodic motions. We have shown that the theoretical transition point is consistent with the simulation result. Finally, we have shown that the dynamical system of cyclic trams in a city is simple but exhibits very complex behavior. In the result, one is not able to predict, deterministically, the arrival time of the tram at the stop but can obtain the probabilistic estimate on the arrival time. We have found the well-controlled operation condition for the cyclic tram system. References [1] [2] [3] [4] [5] [6] [7]
D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. D. Chowdhury, L. Santen, A. Schadschneider, Phys. Rep. 329 (2000) 199. B.S. Kerner, Phys. Rev. E 65 (2002) 046138. K. Nagel, J. Esser, M. Rickert, Annu. Rev. Comput. Phys. 7 (2000) 151. T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331. A. Schadschneider, Physica A 313 (2002) 153. D.E. Wolf, M. Schreckenberg, A. Bachem (Eds.), Tra;c and Granular Flow, World ScientiDc, Singapore, 1996. [8] M. Schreckenberg, D.E. Wolf (Eds.), Tra;c and Granular Flow ’97, Springer, New York, 1998. [9] D. Helbing, H.J. Hermann, M. Schreckenberg, D.E. Wolf (Eds.), Tra;c and Granular Flow ’99, Springer, New York, 2000. [10] O.J. O’loan, M.R. Evans, M.E. Cates, Phys. Rev. E 58 (1998) 1404.
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T. Nagatani / Physica A 331 (2004) 279 – 290 D. Chowdhury, R.C. Desai, Eur. Phys. J. B 15 (2000) 375. T. Nagatani, Physica A 287 (2000) 302. T. Nagatani, Phys. Rev. E 63 (2001) 036116. T. Nagatani, Physica A 296 (2001) 320. T. Nagatani, Physica A 305 (2002) 629. H.J.C. Huijberts, Physica A 308 (2002) 489. S.A. Hill, Physica A (2003), cond-mat/0206008 (2002). T. Nagatani, Physica A 297 (2001) 260. T. Nagatani, Phys. Rev. E 66 (2002) 046103. T. Nagatani, Physica A 319 (2003) 568. T. Nagatani, Physica A 323 (2003) 686.
Physica A 331 (2004) 279 – 290
www.elsevier.com/locate/physa
Fluctuation of tour time induced by interactions between cyclic trams Takashi Nagatani∗ Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan
Abstract We study the dynamical behavior of N trams which move around the tram stops on a cyclic route repeatedly. We present the dynamical model for the cyclic trams. When a tram catches another tram, the tram restarts after delay time Tmin and keeps the minimal time headway Tmin . The distinct dynamical states (the regular, periodic, and chaotic motions) are found by varying loading parameter , delay time Tmin , and number M of tram stops. It is shown that the dynamical transitions occur from the regular motion, through multiply periodic motions, and to the chaotic motion. In the periodic and chaotic motions, the tour and arrival times of trams +uctuate highly. It is shown that the tram-stop’s number M has an important e.ect on the tour time of trams. The phase diagram (region map) is found. c 2003 Elsevier B.V. All rights reserved. PACS: 89.40.+k; 05.70.Fh; 05.90.+m Keywords: Dynamical transition; Chaos; Tram; Phase diagram
1. Introduction Recently, the transportation problems have attracted much attention [1–9]. The vehicular tra;c +ow has been investigated by various models [1–6]. There have been some corresponding studies of buses [10–17]. The bus-route system is a typical dynamical system of interacting buses and passengers. The dynamics of a bus route is di.erent from the tra;c dynamics since buses interact with passengers at designated bus stops in addition to the interaction between buses. ∗
Fax: +81-53-478-1048. E-mail address: [email protected] (T. Nagatani).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.07.007
280
T. Nagatani / Physica A 331 (2004) 279 – 290
In the system with many buses, the dynamical phase transition to the inhomogeneous jammed (bunching) phase has been found [10–17]. If a bus is delayed by some +uctuation, the time headway (gap) between itself and its predecessor becomes larger than the initial time headway since it has to pick up more passengers. The slowly moving delayed bus slows down the buses behind it. As a result, all buses behind it bunch together. The single cyclic bus system also exhibits the nontrivial delay transition [18,19]. In a single cyclic bus system with many bus stops, the bus interacts with prospective passengers waiting at bus stops. This interaction induces the dynamical phase transition between the delay and scheduled time phases. The delay transition is fundamentally di.erent from the bunching transition of many buses. The single cyclic bus model has been extended to the shuttle-bus system with a few buses [20,21]. The buses shuttle between a starting point and a destination on a route. There are the two interactions: the one is the interaction between buses and the other is the interaction between a shuttle bus and passengers waiting at the starting point. It has been found that the two interactions induce a complex behavior of buses. The shuttle-bus system exhibits the dynamical transitions to the chaotic and periodic motions. Also, the motions of trams (streetcars) are similar to those of shuttle buses. The trams move around the stops repeatedly. The passengers waiting at a tram stop board the coming tram. Generally, the perspective passengers waiting at a tram stop increases accordingly as the time headway between a tram and the next tram increases. The tram cannot pass over the other tram. It will be expected that the motions of trams exhibit a complex behavior by the interaction between trams and passengers. The complex motions of trams will induce the large +uctuation of arrival time and riding passengers. It is very important to estimate the arrival time and time headway of trams for public demand. In this paper, we study the dynamical behavior of a few trams on a cyclic route. We present the dynamical model for the cyclic trams. We investigate the +uctuation of arrival time and tour time. We clarify that the trams exhibit the periodic and chaotic motions. We study the dynamical phase transitions among the distinct dynamical states. We present the phase diagram (region map) of the distinct dynamical states. We Dnd the optimal time headway for keeping the well-controlled operation. 2. Model We present the dynamical model of N trams moving on a cyclic route with M tram stops. Trams start at tram stop m = 1, move around the route, stop at all the tram stops m = 1; : : : ; M , and return to tram stop m = 1. All the trams move around the cyclic route repeatedly without passing other trams. The model is deDned on a Dnite one-dimensional lattice with the periodic boundary. Each lattice site is labeled by number m running from 1 to M . A site represents a tram stop. All the trams are numbered from 1 to N in regular order. The distance between a stop m − 1 and its next stop m is deDned by Lm .
T. Nagatani / Physica A 331 (2004) 279 – 290
281
If a tram catches an other tram, its tram stops and restarts after delay time Tmin . The tram does not change the speed except for stopping at the tram stops. The tram keeps constant average speed until it stops. The new passengers arrive at all the tram stops at the rate . So Gt is the number of passengers that have arrived since its predecessor has left the tram stop, where Gt is the time headway between itself and its predecessor. We assume that new passengers try to board the tram after the passengers left the tram. The time ts; m is the amount of time needed to leave the passengers at the tram stop m. The parameter is the time it takes one passenger to board the tram, so Gt is the amount of time needed to board all the passengers at the tram stop. The arrival time t1 (m; n + 1) of tram 1 on tram stop m at trip n + 1 is given by t1 (m; n + 1) = t1 (m; n) +
M
m−1
m=1
k=1
L + ts; m + [t1 (k; n + 1) − tN (k; n)] V
M [t1 (k; n) − tN (k; n − 1)] + 1 (m; n) ; +
(1)
k=m
M where L = m=1 Lm is the length of the route and V is the average speed of trams. 1 (m; n) is a noise of tram 1 at on tram stop m at trip n. The noise is due to a +uctuation of passengers waiting on tram stops. The mean and correlation of the noise are given by 1 (m; n) = 0
and
1 (m; n)i (m ; n ) = a2 1i mm nn :
If tram 1 catches tram N (t1 (m; n + 1) ¡ tN (m; n)) at tram stop m, tram 1 restarts after delay time Tmin : t1 (m; n + 1) = tN (m; n) + Tmin :
(2)
Similarly, the arrival time ti (m; n + 1) of tram i(i ¿ 2) on tram stop m at trip n + 1 is given by ti (m; n + 1) = ti (m; n) +
+
M
M
m−1
m=1
k=1
L ts; m + [ti (k; n + 1) − ti−1 (k; n + 1)] + V
[ti (k; n) − ti−1 (k; n)] + i (m; n) :
(3)
k=m
If tram i catches tram i − 1(ti (m; n + 1) ¡ ti−1 (m; n + 1)), tram i restarts after delay time Tmin : ti (m; n + 1) = ti−1 (m; n + 1) + Tmin :
(4) M
We assume that the amount of time needed to leave the tram is constant m=1 ts; m = M constant. By dividing time by the characteristic time L=V + m=1 ts; m , one obtains the
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governing equations of dimensionless arrival times m−1
T1 (m; n + 1) = T1 (m; n) + 1 +
[T1 (k; n + 1) − TN (k; n)]
k=1
+
M
[T1 (k; n) − TN (k; n − 1)] + 1 (m; n) ;
(5)
k=m
if T1 (m; n + 1) ¡ TN (m; n) ;
T1 (m; n + 1) = TN (m; n) + Tmin Ti (m; n + 1) = Ti (m; n) + 1 +
m−1
(6)
[Ti (k; n + 1) − Ti−1 (k; n + 1)]
k=1
+
M
[Ti (k; n) − Ti−1 (k; n)] + i (m; n) ;
(7)
k=m
Ti (m; n + 1) = Ti−1 (m; n + 1) + Tmin where
Ti (n) ≡ ti (n)
L=V +
L=V +
Tmin ≡ tmin
M
ts; m
a =a
L=V +
L=V + M m=1
; M
ts; m
;
m=1
ts; m
;
ts; m
m=1 M
= i (m; n)
(8)
m=1
i (m; n)
if Ti (m; n + 1) ¡ Ti−1 (m; n + 1) ;
;
2
i (m; n)j (m ; n ) = a ij mm nn : The tram’s motion depends on parameters , Tmin , number M of tram stops, and tram’s number N . With increasing loading parameter , the tram slows down since more passengers board the tram. With decreasing delay time Tmin , the following tram becomes faster since less passengers board the tram. Thus, a tram interacts other trams. It is expected that the trams exhibit a complex behavior and dynamical transitions. 3. Simulation result We study the tram’s behavior by the use of iterates of Eqs. (5)–(8). First, we study the dynamical behavior of two trams without noises. The initial condition is given by T1 (0) = 0;
T2 (0) = 0:5;
T1 (1) = 1:0 and T2 (1) = 1:5 :
(9)
Fig. 1 shows the time evolutions of the tour time (one period) GT 1(n) and the time headway GT 12(n) for two values of delay time Tmin under the constant value of
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Fig. 1. Time evolutions of the tour time (one period) GT 1(n) and the time headway GT 12(n) for two values of delay time Tmin under the constant value of loading parameter where GT 1(n) ≡ T1 (n) − T1 (n) − T1 (n − 1) and GT 12(n) ≡ T1 (n). Diagrams (a) and (b) are obtained for Tmin = 0:4 and 1.0 where = 0:15, tram’s number N = 2, and stop’s number M = 4.
loading parameter where tram stop’s number M = 4, GT 1(n) ≡ T1 (n) − T1 (n − 1), and GT 12(n) ≡ T2 (n) − T1 (n). Diagrams (a) and (b) are obtained for Tmin = 0:4 and 1.0 where = 0:15. For (a) Tmin = 0:4, two trams move with a constant value of tour time and tram 2 always follows tram 1 with constant value Tmin of time headway. For (b) Tmin = 1:0, the motions of two trams exhibit an irregular oscillation. The oscillation may exhibit a chaotic behavior. The tour time and time headway oscillate largely and irregularly. We calculate the variation of tour time GT 1(n) by increasing return n to study the dynamical transitions to the complex motions. Fig. 2 shows the plot of the values of tour time GT 1(n) against delay time Tmin from su;ciently large return n = 100 to 300 where tram’s number N = 2 and = 0:15. Diagrams (a) – (c) are obtained for tram-stop’s number M = 2– 4. In the case of (a) M = 2, when delay time Tmin is less than the critical value Tmin; c = 0:247, the tour time converges to the constant value. Tram 2 follows tram 1 with the constant time headway Tmin . If delay time Tmin is higher than the critical value Tmin; c = 0:247, the tour time of trams oscillate periodically. The dynamical transition from the regular motion to the periodic motion occurs at Tmin; c =0:247. The tour time exhibits the complex behavior with increasing the delay time Tmin . For small values of tour time, the tour time decreases accordingly as the delay time increases. However, the tour time increases with the delay time when the delay time is higher than about one. In the case of (b) M = 3, when delay time Tmin is less than the critical value Tmin; c = 0:361, the tour time converges to the constant value. The dynamical transition from the regular motion to the periodic motion occurs at Tmin; c = 0:361. The transition point is higher than that in Fig. 2(a) of M = 2. The +uctuation of the tour time is larger than that in Fig. 2(a). The dynamical behavior is qualitatively similar to that in Fig. 2(a) but is quantitatively di.erent from that in Fig. 2(a). In the case of (c) M = 4, when delay time Tmin is less than the critical value Tmin; c = 0:484, the tour time converges to the constant value. The dynamical transition
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Fig. 2. Plot of the values of tour time GT 1(n) against delay time Tmin from su;ciently large return n = 100 to 300 where = 0:15 and tram’s number N = 2. (a) Tram-stop’s number M = 2. (b) Tram-stop’s number M = 3. (c) Tram-stop’s number M = 4.
from the regular motion to the periodic motion occurs at Tmin; c = 0:484. The transition point is higher than those in Fig. 2(a) and (b) of M = 2 and 3. The +uctuation of the tour time is larger than those in Fig. 2(a) and (b). The dynamical behavior is qualitatively similar to those in Fig. 2(a) and (b) but is quantitatively di.erent from those in Fig. 2(a) and (b). We calculate the Liapunov exponent to study the chaotic motion. Fig. 3(a) – (c) shows the plot of Liapunov exponent against delay time Tmin for the same values of parameters in Fig. 2(a) – (c). In Fig. 3(a) of tram-stop’s number M = 2, the value of Liapunov exponent changes from a negative value to a positive value at Tmin; c =0:585 and from a positive value to a negative value at Tmin = 0:813. Therefore, the irregular motion in the region between 0:585 ¡ Tmin ¡ 0:813 exhibits the chaos. In Fig. 3(b) of tram-stop’s number M = 3, the value of Liapunov exponent changes from a negative value to a positive value at Tmin; c = 0:642 and from a positive value to a negative value at Tmin = 0:996. Therefore, the irregular motion in the region between 0:642 ¡ Tmin ¡ 0:996 exhibits the chaos. In Fig. 3(c) of tram-stop’s number M = 4, the value of Liapunov exponent changes from a negative value to a positive value at Tmin; c = 0:708 and from a positive value to a negative value at Tmin = 1:233. Therefore, the irregular motion in the region between 0:708 ¡ Tmin ¡ 1:233 exhibits the chaos.
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Fig. 3. Plot of Liapunov exponent against delay time Tmin where = 0:15 and tram’s number N = 2. Plots (a) – (c) correspond, respectively, to those in Fig. 2(a) – (c). (a) Tram-stop’s number M = 2. (b) Tram-stop’s number M = 3. (c) Tram-stop’s number M = 4.
Fig. 4. (a) Enlargement of Fig. 2(b) for 0:35 ¡ Tmin ¡ 0:60. (b) Enlargement of Fig. 2(c) for 0:45 ¡ Tmin ¡ 0:70.
The transition point to chaos increases with the tram-stop’s number. Also, the region of the chaotic motion increases with the tram-stop’s number. Fig. 4(a) shows the enlargement of Fig. 2(b) between 0:35 ¡ Tmin ¡ 0:60. Many transitions occur among periodic motions of di.erent periods. Fig. 4(b) shows the enlargement of Fig. 2(c) between 0:45 ¡ Tmin ¡ 0:70. The Dne structure of periodic motions in the tour times is deDnitely di.erent from that in Fig. 4(a).
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Fig. 5. Plots of average and root-mean-square (rms) of tour time against delay time Tmin for tram-stop’s number M = 4 where = 0:15 and tram’s number N = 2. Point 1 indicates the transition point between the regular and periodic motions. Point 2 indicates the transition point between the periodic and chaotic motions. The chaotic motion appears in the region between points 2 and 3. Point 4 indicates the minimum value of the tour time.
Fig. 6. Plot of the values of tour time GT 1(n) against loading parameter from su;ciently large return n = 100 to 300 under the constant value Tmin = 0:5 of delay time for stop’s number M = 6.
Fig. 5 shows the plots of the average value and root-mean-square (rms) of tour time against delay time Tmin for Fig. 2(c). The mean tour time decreases with increasing the delay time until the transition point 1 at which the regular motion changes the periodic motion. In the periodic motion, the mean tour time increases slightly with the delay time. Then, the average value decreases discontinuously at transition point 2 where the dynamical transition occurs to the chaotic motion. Also, there is the gap in the tour time at the transition point 3 where the chaotic motion changes the periodic motion again. The tour time takes the minimal value of the average at the point 4 where the trams move chaotically. The rms increases linearly with the delay time until point 5. At point 5 where the motion of trams is in the chaotic state, the rms increases abruptly and there is a gap in the rms. Thus, the mean value exhibits the complex behavior due to the periodic and chaotic motions. Fig. 6 shows the plot of the values of tour time GT 1(n) against loading parameter from su;ciently large return n = 100 to 300 under the constant value Tmin = 0:5 of
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Fig. 7. Phase diagram (region map) in phase space (Tmin ; M ) for two trams and loading parameter = 0:15 where Tmin is the delay time and M is the number of tram stops. The regular motion in which the tour time converges to a constant value is indicated by region 1. The periodic motion appears in region 2. The chaotic motion appears in region 3.
Fig. 8. E.ect of noises on the tour time. Plot of tour time GT 1(n) against delay time Tmin from su;ciently large return n = 100 to 300 where = 0:15, tram’s number N = 2, and stop’s number M = 4 at noise’s amplitude a = 0:05. Compare this with Fig. 2(c) of no noise.
delay time for stop’s number M = 6. The tour time increases with loading parameter according as the multiple transitions occur among various periods. We study the phase diagram (region map) for the distinct dynamical states and the dynamical transitions. Fig. 7 shows the phase diagram (region map) in phase space (Tmin ; M ) for two trams and loading parameter = 0:15 where Tmin is the delay time and M is the number of tram stops. The regular motion in which the tour time converges to a constant value is indicated by region 1. The periodic motion appears in region 2. The chaotic motion appears in region 3. We Dnd that the complex behavior of trams depends highly on delay time Tmin , tram-stop’s number M , and loading parameter . We study the e.ect of noises on the motions of trams. Fig. 8 shows the plot of tour time GT 1(n) against delay time Tmin from su;ciently large return n=100 to 300 where =0:15, tram’s number N =2, and stop’s number M =4 at noise’s amplitude a =0:05.
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Compare this with Fig. 2(c) of no noise. The trams exhibit the stochastic motions over a wide range by adding the noises to the deterministic model. The periodic motions disappear. However, the average value of tour time changes little. In order to operate the trams in the optimal state, it is necessary to be in the regular state just below the transition point between the regular and periodic motions. In the regular state just below the transition point, the trams move with the minimal tour time within the region of the regular motion. 4. Analysis We present the theoretical analysis to determine the transition point from the regular motion to the periodic motion. We consider such condition that two trams keep the regular motion. In the regular motion, tram 2 follows tram 1 with keeping constant time headway Tmin . Then, the tour time GT 1 of tram 1 is obtained from Eq. (5) 1 − MTmin : (10) GT 1 = 1 − M The tour time of tram 2 is given by GT 2 = 1 + (1 + M)Tmin :
(11)
If tour time GT 1 of tram 1 is higher than GT 2 of tram 2, tram 2 always catches tram 1 and tram 2 follows tram 1 with keeping time headway Tmin . Thus, one obtains the following condition to maintain the regular motion: M : (12) Tmin ¡ 1 + M − (M)2 The transition point Tmin; c from the regular motion to the periodic motion is given by M : (13) Tmin; c = 1 + M − (M)2 Fig. 9 shows the plot of transition point against tram-stop’s number M for = 0:09; 0:12, and 0.15. The transition point increases accordingly as stop’s number M increases under the condition of constant loading parameter. The solid curve represents
Fig. 9. Plot of transition point between the regular and periodic motions against tram-stop’s number M for loading parameter = 0:09; 0:12, and 0.15 where N = 2. The circles indicate the simulation result. The solid lines represent the theoretical result.
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the theoretical transition point (13). The circular point indicates the simulation result. The theoretical result agrees with simulation result. When M = 1, Eq. (13) agrees with that obtained for two shuttle buses [20]. In Eq. (10), the tour time diverges when loading parameter approaches to 1=M . Also, under the condition of constant loading parameter, the tour time diverges when stop’s number M is higher than 1=. Thus, one obtains the following relation for Dnite tour time: 1 M¡ : (14) We showed that the tour time diverges when stop’s number M is equal to or higher than 7 for = 0:15 in the simulation of Section 3. This simulation result is consistent with Eq. (14). 5. Summary We have studied the dynamical behavior of trams moving around the tram stops repeatedly. We have presented the dynamical model of N trams, which is described by four parameters: loading parameter , minimum headway Tmin , tram-stop’s number M , and tram’s number N . We have found that there are the distinct dynamical states: the regular motion, the periodic motion, and the chaotic motion. We have shown that the dynamical transitions occur among the distinct dynamical states with varying their parameters. Especially, we have shown that the number of tram stops has the important e.ect on the motion of trams. We have presented the theoretical result to predict the transition point between the regular and periodic motions. We have shown that the theoretical transition point is consistent with the simulation result. Finally, we have shown that the dynamical system of cyclic trams in a city is simple but exhibits very complex behavior. In the result, one is not able to predict, deterministically, the arrival time of the tram at the stop but can obtain the probabilistic estimate on the arrival time. We have found the well-controlled operation condition for the cyclic tram system. References [1] [2] [3] [4] [5] [6] [7]
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