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Network traffic flow is an aggregated result of a huge number of travelers' route choices, which is influenced by the travelers' choice behaviors. So day-to-day traffic flow is not static, but presents a complex and tortuous day-to-day dynamic evolution process. Studying day-to-day dynamic evolution of network traffic flow, we can not only know whether the traffic network equilibrium can be reached and how the process is achieved, but also can know what phenomenon will occur in the evolution of network traffic flow if the equilibrium is not reached. In a real traffic system, taking day as scale unit, the day-to-day network traffic demand is variable and changes with everyday's traffic network state. The travelers' route choices are also influenced by the previous day's behaviors and network state. Then, will the day-to-day network traffic flow evolution be stable? If it is unstable, when will bifurcation and chaos occur? In this paper we discuss the day-to-day dynamic evolution of network traffic flow with elastic demand in a simple two-route network. The dynamic evolution model of network traffic flow with elastic demand is formulated. Based on a nonlinear dynamic theory, the existence and uniqueness of the fixed point of dynamic evolution model are proved, and an equilibrium stability condition for the dynamic evolution of network traffic flow with elastic demand is derived. Then, the evolution of network traffic flow is investigated through numerical experiments by changing the three parameters associated with travelers, which are the sensitivity of travelers' travel demand to travel cost, the randomness of travelers' route choices, and travelers' reliance on the previous day's actual cost. Our findings are as follows. Firstly, there are three kinds of final states in the evolution of network traffic flow: stability and convergence to equilibrium, periodic motion and chaos. The final state of the network traffic flow evolution is related to the above three parameters. It is found that under certain conditions the bifurcation diagram of the network traffic flow evolution reveals a complicated phenomenon of period doubling bifurcation to chaos, and then period-halving bifurcation. Meanwhile, the chaotic region is interspersed with odd periodic windows. Moreover, the more sensitive to cost the travelers' travel demand the more likely the system evolution is to be stable. The smaller the randomness of travelers' route choices, the less likely the system evolution is to be stable. The lower the degree of travelers' reliance on the previous day's actual cost, the more likely the system evolution is to be stable.
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Keywords:
- network traffic flow /
- elastic demand /
- dynamical evolution /
- chaos
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[1] Liu S X, Guan H Z, Yan H 2012 Acta Phys. Sin. 61 090506 (in Chinese) [刘诗序, 关宏志, 严海 2012 物理学报 61 090506]
[2] Sheffi Y 1985 Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods (Englewood Cliffs: Prentice-Hall, Inc.) pp59-61, 309
[3] Kim H, Oh J S, Jayakrishnan R 2009 KSCE J. Civil Engineer. 13 117
[4] Nakayama S, Kitamura R, Fujii S 1999 Transp. Res. Rec. 1676 30
[5] Klgl F, Bazzan A L C 2004 J. Artificial Societies and Social Simulation 7 1
[6] Wei F F, Ma S F, Jia N 2014 Math. Probl. Eng. 204 1
[7] Kusakabe T, Nakano Y 2015 Transp. Res. C 59 278
[8] Liu T L, Huang H J 2007 Acta Phys. Sin. 56 6321 (in Chinese) [刘天亮, 黄海军 2007 物理学报 56 6321]
[9] Liu S X, Guan H Z 2013 China Civil Eng. J. 46 136 (in Chinese) [刘诗序, 关宏志 2013 土木工程学报 46 136]
[10] Iida Y, Akiyama T, Uchida T 1992 Transp. Res. B 26 17
[11] Selten R, Chmura T, Pitz T, Kubec S, Schreckenberg M 2007 Game Econ. Behav. 58 394
[12] Rapoport A, Gisches E J, Daniel T 2014 Transp. Res. B 68 154
[13] Smith M J 1984 Transp. Sci. 18 245
[14] Smith M J, Watling D P 2016 Transp. Res. B 85 132
[15] Nagurney A, Zhang D 1997 Transp. Sci. 31 147
[16] Watling D 1999 Transp. Res. B 33 281
[17] Cho H J, Hwang M C 2005 Math. Comput. Model. 41 501
[18] Kumar A, Peeta S 2015 Transp. Res. B 80 235
[19] Tan Z, Yang H, Guo R Y 2015 Transp. Res. C 61 87
[20] Di X, Liu H X, Ban X X, Yu J W 2015 Netw. Spat. Econ. 15 537
[21] He X Z, Peeta S 2016 Transp. Res. B 84 237
[22] Iryo T 2016 Transp. Res. B 92 88
[23] Xiao F, Yang H, Ye H B 2016 Transp. Res. B 86 86
[24] Guo R Y, Huang H J 2008 J. Manag. Sci. in China 11 12 (in Chinese) [郭仁拥, 黄海军 2008 管理科学学报 11 12]
[25] Guo R Y, Yang H, Huang H J, Tan Z J 2015 Transp. Res. B 71 248
[26] Zhang B, Juan Z C, Ni A N 2014 J. Ind. Eng. Eng. Manag. 28 164
[27] Horowitz J L 1984 Transp. Res. B 18 13
[28] Cascetta E, Cantarella G E 1991 Transp. Res. A 25 277
[29] Cantarella G E, Cascetta E 1995 Transp. Sci. 29 305
[30] Cantarella G E 2013 Transp. Res. C 29 117
[31] Watling D, Hazelton M L 2003 Netw. Spat. Econ. 3 349
[32] Bie J, Lo H K 2010 Transp. Res. B 44 90
[33] He X, Liu H X 2012 Transp. Res. B 46 50
[34] Han L, Du L 2012 Transp. Res. B 46 72
[35] Zhao X, Orosz G 2014 Physica D 275 54
[36] Di X, Liu H X 2016 Transp. Res. B 85 142
[37] Li T, Guan H Z, Liang K K 2016 Acta Phys. Sin. 65 150502 (in Chinese) [李涛, 关宏志, 梁科科 2016 物理学报 65 150502]
[38] Guo R Y, Yang H, Huang H J 2013 Transp. Res. C 34 121
[39] Guo R Y, Huang H J 2016 Transp. Res. C 71 122
[40] Yang W J, Guo R Y, Li Q 2015 Syst. Eng. Theory Pract. 35 3192 (in Chinese) [杨文娟, 郭仁拥, 李琦 2015 系统工程理论与实践 35 3192]
[41] Xu H L, Yu X L, Zhou J 2015 J. Manag. Sci. China 18 39 (in Chinese) [徐红利, 于新莲, 周晶 2015 管理科学学报 18 39]
[42] Cantarella G E, Watling D P 2016 Euro. J. Transp. Logist. 5 69
[43] Dafermos S 1982 Networks 12 57
[44] Cantarella G E 1997 Transp. Sci. 31 107
[45] Yu Q, Fang D B, Du W 2014 Eur. J. Oper. Res. 239 112
[46] Zhou J 2001 J. Syst. Engineer. 16 88 (in Chinese) [周晶 2001 系统工程学报 16 88]
[47] Nagurney A 1999 Network Economics: A Variational Inequality Approach (Boston: Kluwer Academic Publishers) pp17-19
[48] Liu Z H 2006 Fundamentals and Applications of Chaotic Dynamic (Beijing: Higher Education Press) pp24-29, 60-61 (in Chinese) [刘宗华 2006 混沌动力学基础及其应用 (北京: 高等教育出版社) 第2429, 6061页]
[49] Stone L 1993 Nature 365 617
[50] Yu W B, Wei X P 2006 Acta Phys. Sin. 55 3969 (in Chinese) [于万波, 魏小鹏 2006 物理学报 55 3969]
[51] Peng M S 2005 Chaos Solitons Fract. 25 1123
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