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.

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.
Y.
Bar-Yam
,
Dynamics of Complex Systems
(
Addison-Wesley
,
1997
).
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
).
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