We propose a model of a bus traveling through a sequence of traffic lights, which is required to stop between the traffic signals to pick up passengers. A two dimensional model, of velocity and traveled time at each traffic light, is constructed, which shows non-trivial and chaotic behaviors for realistic city traffic parameters. We restrict the parameter values where these non-trivial and chaotic behaviors occur, by following analytically and numerically the fixed points and period 2 orbits. We define conditions where chaos may arise by determining regions in parameter space where the maximum Lyapunov exponent is positive. Chaos seems to occur as long as the ratio of the braking and accelerating capacities are greater than about ∼3.
References
1.
D.
Helbing
and M.
Treiber
, “Jams, waves, and clusters
,” Science
282
, 2001
–2003
(1998
).2.
B. A.
Toledo
, V.
Muñoz
, J.
Rogan
, C.
Tenreiro
, and J. A.
Valdivia
, “Modeling traffic through a sequence of traffic lights
,” Phys. Rev. E
70
(1
), 016107
(2004
).3.
B. A.
Toledo
, E.
Cerda
, J.
Rogan
, V.
Muñoz
, C.
Tenreiro
, R.
Zarama
, and J. A.
Valdivia
, “Universal and nonuniversal features in a model of city traffic
,” Phys. Rev. E
75
, 026108
(2007
).4.
L. A.
Wastavino
, B. A.
Toledo
, J.
Rogan
, R.
Zarama
, V.
Muñoz
, and J. A.
Valdivia
, “Modeling traffic on crossroads
,” Physica A
381
(0
), 411
–419
(2007
).5.
T.
Nagatani
, “The physics of traffic jams
,” Rep. Prog. Phys.
65
(9
), 1331
–1386
(2002
).6.
T.
Nagatani
, “Clustering and maximal flow in vehicular traffic through a sequence of traffic lights
,” Phys. A
377
(2
), 651
–660
(2007
).7.
S.
Lammer
and D.
Helbing
, “Self-control of traffic lights and vehicle flows in urban road networks
,” J. Stat. Mech.: Theory Exp.
2008
, 04019
.8.
J.
Villalobos
, B. A.
Toledo
, D.
Pastén
, V.
Muñoz
, J.
Rogan
, R.
Zarama
, N.
Lammoglia
, and J. A.
Valdivia
, “Characterization of the nontrivial and chaotic behavior that occurs in a simple city traffic model
,” Chaos
20
, 013109
(2010
).9.
B. A.
Toledo
, M. A. F.
Sanjuan
, V.
Muñoz
, J.
Rogan
, and J. A.
Valdivia
, “Non-smooth transitions in a simple city traffic model analyzed through supertracks
,” Commun. Nonlinear Sci. Numer. Simul.
18
(1
), 81
–88
(2013
).10.
D.
Pastén
, V.
Muñoz
, B.
Toledo
, J.
Villalobos
, R.
Zarama
, J.
Rogan
, and J. A.
Valdivia
, “Universal behavior in a model of city traffic with unequal green/red times
,” Physica A
391
(21
), 5230
–5243
(2012
).11.
J.
Villalobos
, V.
Muñoz
, J.
Rogan
, R.
Zarama
, N. F.
Johnson
, B.
Toledo
, and J. A.
Valdivia
, “Regular transport dynamics produce chaotic travel times
,” Phys. Rev. E
89
, 062922
(2014
).12.
13.
Y.
Bar-Yam
, Unifying Themes in Complex Systems
, New England Complex Systems Institute Series on Complexity (Westview Press
, 2003
).14.
G.
Nicolis
and I.
Prigogine
, Exploring Complexity: An Introduction
(W. H. Freeman & Company
, 1989
).15.
H. K.
Lee
, H.-W.
Lee
, and D.
Kim
, “Macroscopic traffic models from microscopic car-following models
,” Phys. Rev. E
64
(5
), 056126
(2001
).16.
J.
Matsukidaira
and K.
Nishinari
, “Euler-Lagrange correspondence of cellular automaton for traffic-flow models
,” Phys. Rev. Lett.
90
(8
), 088701
(2003
).17.
N.
Mitarai
and H.
Nakanishi
, “Spatiotemporal structure of traffic flow in a system with an open boundary
,” Phys. Rev. Lett.
85
(8
), 1766
–1769
(2000
).18.
K.
Nishinari
, M.
Treiber
, and D.
Helbing
, “Interpreting the wide scattering of synchronized traffic data by time gap statistics
,” Phys. Rev. E
68
(6
), 067101
(2003
).19.
Z.
Su
, L.
Li
, H.
Peng
, J.
Kurths
, J.
Xiao
, and Y.
Yang
, “Robustness of interrelated traffic networks to cascading failures
,” Sci. Rep.
4
, 2045
–2322
(2014
).20.
H. K.
Lee
, R.
Barlovic
, M.
Schreckenberg
, and D.
Kim
, “Mechanical restriction versus human overreaction triggering congested traffic states
,” Phys. Rev. Lett.
92
(23
), 238702
(2004
).21.
F.
Castillo
, B. A.
Toledo
, V.
Muñoz
, J.
Rogan
, R.
Zarama
, M.
Kiwi
, and J. A.
Valdivia
, “City traffic jam relief by stochastic resonance
,” Physica A
403
, 65
–70
(2013
).22.
S.
Tadaki
, M.
Kikuchi
, A.
Nakayama
, K.
Nishinari
, A.
Shibata
, Y.
Sugiyama
, and S.
Yukawa
, “Power-law fluctuation in expressway traffic flow: Detrended fluctuation analysis
,” J. Phys. Soc. Jpn.
75
(3
), 034002
(2006
).23.
S. E.
Christodoulou
, “Traffic modeling and college-bus routing using entropy maximization
,” J. Transp. Eng.
136
(2
), 102
–109
(2010
).24.
O. H.
Gao
and R. A.
Klein
, “Environmental equity in participation of the clean air school bus program: The case of New York state
,” Transp. Res. Part D: Transport Environ.
15
(4
), 220
–227
(2010
).25.
Jing-Quan
Li
and K.
Larry Head
, “Sustainability provisions in the bus-scheduling problem
,” Transp. Res. Part D: Transport Environ.
14
(1
), 50
–60
(2009
).26.
H.
Li
and R. L.
Bertini
, “Assessing a model for optimal bus stop spacing with high-resolution archived stop-level data
,” Transp. Res. Rec.
2111
, 24
–32
(2009
).27.
J.
Bin
, L.
Xin-Gang
, J.
Rui
, and G.
Zi-You
, “The influence of bus stop on the dynamics of traffic flow
,” Acta Phys. Sin.
58
(10
), 6845
–6851
(2009
).28.
D.
Jian-Xun
and H.
Hai-Jun
, “A public transport system model with consideration of bus stop
,” Acta Phys. Sin.
59
(5
), 3093
–3098
(2010
).29.
Tie-Qiao
Tang
, Y.
Li
, and Hai-Jun
Huang
, “The effects of bus stop on traffic flow
,” Int. J. Mod. Phys. C
20
(6
), 941
–952
(2009
).30.
Á.
Ibeas
, L.
dell'Olio
, B.
Alonso
, and O.
Sainz
, “Optimizing bus stop spacing in urban areas
,” Transp. Res. Part E: Logist. Transp. Rev.
46
(3
), 446
–458
(2010
).31.
M.
Estrada
, C.
Trapote
, M.
Roca-Riu
, and F.
Robusté
, “Improving bus travel times with passive traffic signal coordination
,” Transp. Res. Rec.: J. Transp. Res. Board
2111
, 68
–75
(2009
).32.
Chen-Fu
Liao
and G. A.
Davis
, “Simulation study of bus signal priority strategy—Taking advantage of global positioning system, automated vehicle location system, and wireless communications
,” Transp. Res. Rec.
2034
, 82
–91
(2007
).33.
L. A.
Koehler
and W.
Kraus
, Jr., “Simultaneous control of traffic lights and bus departure for priority operation
,” Transp. Res. Part C: Emerging Technol.
18
(3
), 288
–298
(2010
); in Proceedings of 11th IFAC Symposium: The Role of Control
.34.
M.
Chao-Qun
, H.
Hai-Jun
, T.
Tie-Qiao
, and W.
Hui-Wen
, “Influences of signal light and bus-stop position on T-road junction traffic
,” Acta Phys. Sin.
58
(3
), 1497
–1503
(2009
).35.
N. B.
Hounsell
, B. P.
Shrestha
, F. N.
McLeod
, S.
Palmer
, T.
Bowen
, and J. R.
Head
, “Using global positioning system for bus priority in London: Traffic signals close to bus stops
,” IET Intel. Transport Syst.
1
(2
), 131
–137
(2007
).36.
S.
Yu-Kun
and Z.
Xiao-Mei
, “Combined bottleneck effect of on-ramp and bus stop in a cellular automaton model
,” Chin. Phys. B
18
(12
), 5242
–5248
(2009
).37.
L.
Qing-Ding
, D.
Li-Yun
, and D.
Shi-Qiang
, “Investigation on traffic bottleneck induce by bus stopping with a two-lane cellular automaton model
,” Acta Phys. Sin.
58
(11
), 7584
–7590
(2009
).38.
Q.
Yong-Sheng
, W.
Hai-Long
, and W.
Chun-Lei
, “The study of a cellular automaton traffic flow model with public transit, harbor-shaped bus stop, and mixed different-maximum-speed vehicles on single lane
,” Acta Phys. Sin.
57
(4
), 2115
–2121
(2008
).39.
Y.
Xu-Hua
, S.
Bao
, W.
Bo
, and S.
You-Xian
, “Mean-field theory for some bus transport networks with random overlapping clique structure
,” Commun. Theor. Phys.
53
(4
), 688
(2010
).40.
H.
Chang
, D.
Park
, S.
Lee
, H.
Lee
, and S.
Baek
, “Dynamic multi-interval bus travel time prediction using bus transit data
,” Transportmetrica
6
(1
), 19
–38
(2010
).41.
B.
Yu
, Z. Z.
Yang
, and J.
Wang
, “Bus travel-time prediction based on bus speed
,” Proc. Inst. Civ. Eng. Transp.
163
(1
), 3
–7
(2010
).42.
V.
Thamizh Arasan
and P.
Vedagiri
, “Bus priority on roads carrying heterogeneous traffic: A study using computer simulation
,” Eur. J. Transport Infrastruct. Res.
8
(1
), 45
–63
(2008
).43.
Y.
Hadas
and A. A.
Cedar
, “Improving bus passenger transfers on road segments through online operational tactics
,” Transp. Res. Rec.
2072
, 101
–109
(2008
).44.
Y.
Lao
and L.
Liu
, “Performance evaluation of bus lines with data envelopment analysis and geographic information systems
,” Comput. Environ. Urban Syst.
33
(4
), 247
–255
(2009
).45.
G.
Shen
and X.
Kong
, “Study on road network traffic coordination control technique with bus priority
,” IEEE Trans. Syst. Man Cybern. Part C
39
(3
), 343
–351
(2009
).46.
J. A.
Sorratini
, R.
Liu
, and S.
Sinha
, “Assessing bus transport reliability using micro-simulation
,” Transp. Plann. Technol.
31
(3
), 303
–324
(2008
).47.
N.
Uno
, F.
Kurauchi
, H.
Tamura
, and Y.
Iida
, “Using bus probe data for analysis of travel time variability
,” J. Intell. Transp. Syst.
13
(1
), 2
–15
(2009
).48.
M.-M.
Yu
and C.-K
Fan
, “Measuring the performance of multimode bus transit: A mixed structure network DEA model
,” Transp. Res. Part E: Logist. Transp. Rev.
45
(3
), 501
–515
(2009
).49.
S.
Li
and Y.
Ju
, “Evaluation of bus-exclusive lanes
,” Trans. Intell. Transp. Syst.
10
(2
), 236
–245
(2009
).50.
K.
Sakamoto
, C.
Abhayantha
, and H.
Kubota
, “Effectiveness of bus priority lane as countermeasure for congestion
,” Transp. Res. Rec.
2034
, 103
–111
(2007
).51.
V.
Thamizh Arasan
and P.
Vedagiri
, “Simulating heterogeneous traffic flow on roads with and without bus lanes
,” J. Infrastruct. Syst.
15
(4
), 305
–312
(2009
).52.
T.
Nagatani
, “Dynamics and schedule of shuttle bus controlled by traffic signal
,” Physica A
387
, 5892
–5900
(2008
).53.
A.
Cain
, G.
Darido
, M.
Baltes
, P.
Rodriguez
, and J.
Barrios
, “Applicability of Transmilenio bus rapid transit system of Bogotá, Colombia, to the United States
,” Transp. Res. Rec.: J. Transp. Res. Board
2034
(1
), 45
–54
(2007
).54.
A.
Gilbert
, “Bus rapid transit: Is “Transmilenio” a miracle cure?
,” Transport Rev.: A Transnational Transdiscip. J.
28
(4
), 439
–467
(2008
).55.
D.
Hensher
and T.
Golob
, “Bus rapid transit systems: A comparative assessment
,” Transportation
35
(4
), 501
–518
(2008
).56.
K.
Nagel
and M.
Schreckenberg
, “A cellular automaton model for freeway traffic
,” J. Phys. I
2
(12
), 2221
(1992
).© 2015 AIP Publishing LLC.
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