# Complex Dynamics of Bus, Tram, and Elevator Delays in Transportation Systems

• Takashi Nagatani
Reference work entry
Part of the Encyclopedia of Complexity and Systems Science Series book series (ECSSS)

## Glossary

Combined map model

The combined map is a nonlinear map combined with two nonlinear maps. The combined map model is a dynamic model of a single bus in which both effects of speed control and periodic inflow is taken into account.

Delayed map model

Nonlinear map with a time lag. The delayed map model is a dynamic model of bus motion described by the delayed map where the delay of acceleration or deceleration is taken into account.

Deterministic chaos

The future behavior of dynamics involved no random elements does not make predictable. The behavior is known as deterministic chaos. The chaos occurs in the nonlinear map models of buses, trams, and elevators.

Extended circle map model

The circle map is a one-dimensional map which maps a circle onto itself. A nonlinear map including sin function. The extended circle map model is a dynamic model of a bus with the periodic inflow at a transfer point.

Nonlinear map model

A nonlinear map is a recurrence relation presented by a nonlinear...

## Primary Literature

1. Chowdhury D, Desai RC (2000) Steady-states and kinetics of ordering in bus-route models: connection with the Nagel-Schreckenberg model. Eur Phys J B 15:375–384
2. Dong C, Ma X, Wang B, Sun X (2010) Effects of prediction feedback in multi-route intelligent traffic systems. Phys A 389:3274–3281
3. Gupta AK, Sharma S, Redhu P (2015) Effect of multi-phase optimal velocity function on jamming transition in a lattice hydrodynamic model with passing. Nonlinear Dyn 80:1091–1108
4. He DR, Yeh WJ, Kao YH (1985) Studies of return maps, chaos, and phase-locked states in a current-driven Josephson-junction simulator. Phys Rev B 31:1359–1373
5. Hino Y, Nagatani T (2014) Effect of bottleneck on route choice in two-route traffic system with real-time information. Phys A 395:425–433
6. Hino Y, Nagatani T (2015) Asymmetric effect of route-length difference and bottleneck on route choice in two-route traffic system. Phys A 428:416–425
7. Huijberts HJC (2002) Analysis of a continuous car-following model for a bus route: existence, stability and bifurcations of synchronous motions. Phys A 308:489–517
8. Kerner BS (2004) Three-phase traffic theory and highway capacity. Phys A 333:379–440
9. Kerner BS (2016a) The maximization of the network throughput ensuring free flow conditions in traffic and transportation networks: breakdown minimization (BM) principle versus Wardrop’s equilibria. Eur Phys J B 89:199
10. Kerner BS (2016b) Failure of classical traffic flow theories: stochastic highway capacity and automatic driving. Phys A 450:700–747
11. Kerner BS (2017) Breakdown minimization principle versus Wardrop’s equilibria for dynamic traffic assignment and control in traffic and transportation networks: a critical mini-review. Phys A 466:626–662
12. Komada K, Kojima K, Nagatani T (2011) Vehicular motion in 2D city traffic network with signals controlled by phase shift. Phys A 390:914–928
13. Lämmer S, Gehlsen B, Helbing D (2006) Scaling laws in the spatial structure of urban road networks. Phys A 363:89–95
14. Li X, Fang K, Peng G (2017) A new lattice model of traffic flow with the consideration of the drivers’ aggressive characteristics. Phys A 468:315–321
15. Masoller C, Rosso OA (2011) Quantifying the complexity of the delayed logistic map. Philos Trans R Soc A 369:425–438
16. Nagatani T (2000) Kinetic clustering and jamming transitions in a car following model of bus route. Phys A 287:302–312
17. Nagatani T (2001a) Bunching transition in a time-headway model of bus route. Phys Rev E 63:036115-1-7
18. Nagatani T (2001b) Interaction between buses and passengers on a bus route. Phys A 296:320–330
19. Nagatani T (2001c) Delay transition of a recurrent bus on a circular route. Phys A 297:260–268
20. Nagatani T (2002a) Bunching and delay in bus-route system with a couple of recurrent buses. Phys A 305:629–639
21. Nagatani T (2002b) Dynamical transition to periodic motions of a recurrent bus induced by nonstops. Phys A 312:251–259
22. Nagatani T (2002c) Transition to chaotic motion of a cyclic bus induced by nonstops. Phys A 316:637–648
23. Nagatani T (2002d) Dynamical behavior in the nonlinear-map model of an elevator. Phys A 310:67–77
24. Nagatani T (2002e) Chaotic and periodic motions of a cyclic bus induced by speedup. Phys Rev E 66:046103-1-7
25. Nagatani T (2003a) Chaotic motion of shuttle buses in two-dimensional map model. Chaos Solitons Fractals 18:731–738
26. Nagatani T (2003b) Complex behavior of elevators in peak traffic. Phys A 326:556–566
27. Nagatani T (2003c) Fluctuation of tour time induced by interactions between cyclic trams. Phys A 331:279–290
28. Nagatani T (2003d) Transitions to chaos of a shuttle bus induced by continuous speedup. Phys A 321:641–652
29. Nagatani T (2003e) Complex motions of shuttle buses by speed control. Phys A 322:685–697
30. Nagatani T (2003f) Dynamical transitions to chaotic and periodic motions of two shuttle buses. Phys A 319:568–578
31. Nagatani T (2003g) Chaos and headway distribution of shuttle buses that pass each other freely. Phys A 323:686–694
32. Nagatani T (2003h) Fluctuation of riding passengers induced by chaotic motions of shuttle buses. Phys Rev E 68:036107-1-8
33. Nagatani T (2003i) Dynamical behavior of N shuttle buses not passing each other: chaotic and periodic motions. Phys A 327:570–582
34. Nagatani T (2004) Dynamical transitions in peak elevator traffic. Phys A 333:441–452
35. Nagatani T (2005a) Self-similar behavior of a single vehicle through periodic traffic lights. Phys A 347:673–682
36. Nagatani T (2005b) Chaos and dynamics of cyclic trucking of size two. Int J Bifurcat Chaos 15:4065–4073
37. Nagatani T (2006a) Control of vehicular traffic through a sequence of traffic lights positioned with disordered interval. Phys A 368:560–566
38. Nagatani T (2006b) Chaos control and schedule of shuttle buses. Phys A 371:683–691
39. Nagatani T (2007a) Clustering and maximal flow in vehicular traffic through a sequence of traffic lights. Phys A 377:651–660
40. Nagatani T (2007b) Dynamical model for retrieval of tram schedule. Phys A 377:661–671
41. Nagatani T (2007c) Passenger’s fluctuation and chaos on ferryboats. Phys A 383:613–623
42. Nagatani T (2008) Dynamics and schedule of shuttle bus controlled by traffic signal. Phys A 387:5892–5900
43. Nagatani T (2011a) Complex motion of shuttle buses in the transportation reducing energy consumption. Phys A 390:4494–4501
44. Nagatani T (2011b) Complex motion in nonlinear-map model of elevators in energy-saving traffic. Phys Lett A 375:2047–2050
45. Nagatani T (2012) Delay effect on schedule in shuttle bus transportation controlled by capacity. Phys A 391:3266–3276
46. Nagatani T (2013a) Dynamics in two-elevator traffic system with real-time information. Phys Lett A 377:3296–3299
47. Nagatani T (2013b) Nonlinear-map model for control of airplane. Phys A 392:6545–6553
48. Nagatani T (2013c) Modified circle map model for complex motion induced by a change of shuttle buses. Phys A 392:3392–3401
49. Nagatani T (2013d) Complex motion of elevators in piecewise map model combined with circle map. Phys Lett A 377:2047–2051
50. Nagatani T (2013e) Nonlinear-map model for bus schedule in capacity-controlled transportation. App Math Model 37:1823–1835
51. Nagatani T (2014) Dynamic behavior in two-route bus system with real-time information. Phys A 413:352–360
52. Nagatani T (2015) Complex motion induced by elevator choice in peak traffic. Phys A 436:159–169
53. Nagatani T (2016a) Effect of stopover on motion of two competing elevators in peak traffic. Phys A 444:613–621
54. Nagatani T (2016b) Effect of speedup delay on shuttle bus schedule. Phys A 460:121–130
55. Nagatani T (2016c) Complex motion of a shuttle bus between two terminals with periodic inflows. Phys A 449:254–264
56. Nagatani T (2017) Effect of periodic inflow on speed-controlled shuttle bus. Phys A 469:224–231
57. Nagatani T, Naito Y (2011) Schedule and complex motion of shuttle bus induced by periodic inflow of passengers. Phys Lett A 375:3579–3582
58. Nagatani T, Tobita K (2012) Effect of periodic inflow on elevator traffic. Phys A 391:4397–4405
59. Nagatani T, Yoshimura J (2002) Dynamical transition in a coupled-map lattice model of a recurrent bus. Phys A 316:625–636
60. Naito Y, Nagatani T (2012) Effect of headway and velocity on safety-collision transition induced by lane changing in traffic flow. Phys A 391:1626–1635
61. O’loan OJ, Evans MR, Cates ME (1998) Jamming transition in a homogeneous one-dimensional system: the bus route model. Phys Rev E 58:1404–1421
62. Redhu P, Gupta AK (2016) The role of passing in a two-dimensional network. Nonlinear Dyn 86:389–399
63. Sugiyama Y, Nagatani T (2012) Multiple-vehicle collision induced by a sudden stop. Phys Lett A 376:1803–1806
64. Sugiyama Y, Nagatani T (2013) Multiple-vehicle collision in traffic flow by a sudden slowdown. Phys A 392:1848–1857
65. Tobita K, Nagatani T (2012) Effect of signals on two-route traffic system with real-time information. Phys A 391:6137–6145
66. Tobita K, Nagatani T (2013) Green-wave control of unbalanced two-route traffic system with signals. Phys A 392:5422–5430
67. Toledo BA, Munoz V, Rogan J, Tenreiro C, Valdivia JA (2004) Modeling traffic through a sequence of traffic lights. Phys Rev E 70:016107
68. Toledo BA, Cerda E, Rogan J, Munoz V, Tenreiro C, Zarama R, Valdivia JA (2007) Universal and nonuniversal feature in a model of city traffic. Phys Rev E 75:026108
69. Treiber M, Hennecke A, Helbing D (2000) Congested traffic states in empirical observations and microscopic simulations. Phys Rev E 62:1805
70. Treiber M, Kesting A, Helbing D (2006) Delays, inaccuracies and anticipation in microscopic traffic models. Phys A 360:71–88
71. Villalobos J, Toledo BA, Pasten D, Nunoz V, Rogan J, Zarma R, Valdivia JA (2010) Characterization of the nontrivial and chaotic behavior that occurs in a simple city traffic model. Chaos 20:013109
72. Wahle J, Lucia A, Bazzan C, Klugl F, Schreckenberg M (2000) Decision dynamics in a traffic scenario. Phys A 287:669–681
73. Wastavino LA, Toledo BA, Rogan J, Zarama R, Muñoz V, Valdivia JA (2008) Modeling traffic on crossroads. Phys A 381:411–419

## Books and Reviews

1. Chowdohury D, Santen L, Schadschneider A (2000) Statistical physics of vehicular traffic and some related systems. Phys Rep 329:199–329
2. Helbing D (2001) Traffic and related self-driven many-particle systems. Rev Mod Phys 73:1067–1141
3. Kerner BS (2004) The physics of traffic. Springer, HeidelbergGoogle Scholar
4. Nagatani T (2002) The physics of traffic jams. Rep Prog Phys 63:1331–1386Google Scholar
5. Paterson SE, Allan LK (2009) Road traffic: safety, modeling, and impacts. Nova Science Publishers, New YorkGoogle Scholar