Chaotic behavior in the dynamical evolution of network traffic flow and its control

Liu Shi-Xu; Guan Hong-Zhi; Yan Hai

doi:10.7498/aps.61.090506

Abstract

This paper presents the day-to-day dynamic evolution of network traffic flow in a simple two-route network. Firstly, a day-to-day dynamical assignment model is formulated, which can depict the evolution of network traffic flow. We have proved that the fixed point of the dynamical system, which is the stochastic user equilibrium solution, exists and is unique. Secondly, based on nonlinear dynamics theory, an equilibrium stability condition for the network is derived. Moreover, the evolution of network traffic flow is investigated through numerical experiments. Meanwhile, periodic and chaotic flows are discovered under certain conditions. Finally, a chaotic control method is derived considering OD demand as control variable.

Keywords:

Authors and contacts

1.
Beijing Key Laboratory of Transportation Engineering, Beijing University of Technology, Beijing 100124, China
Funds: Project supported by the National Basic Research Program of China (Grant No. 2012CB725403), the National Natural Science Foundation of China (Grant No. 50978008), and the Natural Science Foundation of Beijing, China (Grant No. 8102007).

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[29]	Nakayama S 2004 The 83rd Annual Meeting of the Transportation Research Board Washington, D.C., January 11---15, 2004
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[31]	Liu F, Guan Z H, Wang H 2008 Chin. Phys. B 17 2405
[32]	Zhang J F, Pei Q Y, Zhang X L 2011 Chin. Phys. B 20 080503
[33]	Wang J W, Ma Q H, Zeng L 2011 Chin. Phys. B 20 080506
[34]	Li N, Sun H Y, Zhang Q L 2012 Chin. Phys. B 21 010503
[35]	Wang B H 1986 J. Univ. Sci. Technol. China 16 257 (in Chinese) [汪秉宏 1986 中国科学技术大学学报 16 257]
[36]	Stone L 1993 Nature 365 617
[37]	Ruxton G D, Rohani P 1998 Theor. Popul. Biol 53 175
[38]	Stone L, Hart D 1999 Theor. Popul. Biol 55 227
[39]	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]
[40]	Vandermeer J 1997 Ecol. Model. 95 311
[41]	Peng M S 2005 Chaos Solitons Fract. 24 1279
[42]	Peng MS 2005 Chaos Solitons Fract. 25 1123
[43]	Li T Y, Yorke J A 1975 Amer. Math. Monthly 82 985
[44]	Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[45]	Hu G, Xiao J H, Zheng Z G 2000 Chaos Control (Shanghai: Shanghai Scientific and Technological Education Publishing House) p8--20 (in Chinese) [胡岗, 萧井华, 郑志刚 2000 混沌控制(上海: 上海科技教育出版社) 第8---20页]

Cited By

[1]	Wardrop J G 1952 Proceedings of the Institute of Civil Engineers U.K., January 24, 1952 p325
[2]	Sheffi Y 1985 Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.) p22
[3]	Daganzo C, Sheffi Y 1977 Transp. Sci. 11 253
[4]	Vythoulkas P C 1990 Transp. Res. B 24 453
[5]	Leurent F 1993 Eur. J. Oper. Res. 71 205
[6]	Bell M G H 1995 Transp. Res. B 29 125
[7]	Nakayama S, Kitamura R, Fujii S 1999 Transport. Res. Rec. 1676 30
[8]	Nakayama S, Kitamura R 2000 Transport. Res. Rec. 1725 63
[9]	Nakayama S, Kitamura R, Fujii S 2001 Transport. Res. Rec. 1752 62
[10]	Klügl F, Bazzan A L C 2002 Proc. of the First Int. Joing Conf. on Autonomous Agents and Multi-agent Systems Bologna, Italy, July 15--19, 2002 p217
[11]	Klügl F, Bazzan A L C 2004 J. Artificial Societies and Social Simulation 7 1
[12]	Klügl F, Bazzan A L C 2004 J. Intelligent Transportation Systems 8 223
[13]	Liu T L, Huang H J 2005 Intelligent Transportation System 4 17 (in Chinese) [刘天亮, 黄海军 2005 ITS 通讯 4 17]
[14]	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]
[15]	Kim H, Oh J S, Jayakrishnan R 2009 KSCE J. Civil Engineering 13 117
[16]	Smith M J 1984 Transp. Sci. 18 245
[17]	Friesz T L, Bernstein D, Mehta N J, Tobin R L, Ganjalizadeh S 1994 Oper. Res. 42 1120
[18]	Zhang D, Nagurney A 1996 Transp. Res. B 30 245
[19]	Nagurney A, Zhang D 1997 Transp. Sci. 31 147
[20]	Watling D 1999 Transp. Res. B 33 281
[21]	Cho H J, Hwang M C 2005 Math. Comput. Model. 41 501
[22]	Mounce R 2006 Transp. Res. B 40 779
[23]	Guo R Y, Huang H J 2008 J. Management Sciences in China 11 12 (in Chinese) [郭仁拥, 黄海军 2008 管理科学学报 11 12]
[24]	Horowitz J L 1984 Transp. Res. B 18 13
[25]	Cascetta E, Cantarella G E 1991 Transp. Res. A 25 277
[26]	Cantarella G E, Cascetta E 1995 Transp. Sci. 29 305
[27]	Watling D, Hazelton M L 2003 Netw. Spat. Econ. 3 349
[28]	Bie J, Lo H K 2010 Transp. Res. B 44 90
[29]	Nakayama S 2004 The 83rd Annual Meeting of the Transportation Research Board Washington, D.C., January 11---15, 2004
[30]	Liu Z H 2006 Fundamentals and Applications of Chaotic Dynamic (Beijing: Higher Education Press) p9-14, 60 (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9---14, 60页]
[31]	Liu F, Guan Z H, Wang H 2008 Chin. Phys. B 17 2405
[32]	Zhang J F, Pei Q Y, Zhang X L 2011 Chin. Phys. B 20 080503
[33]	Wang J W, Ma Q H, Zeng L 2011 Chin. Phys. B 20 080506
[34]	Li N, Sun H Y, Zhang Q L 2012 Chin. Phys. B 21 010503
[35]	Wang B H 1986 J. Univ. Sci. Technol. China 16 257 (in Chinese) [汪秉宏 1986 中国科学技术大学学报 16 257]
[36]	Stone L 1993 Nature 365 617
[37]	Ruxton G D, Rohani P 1998 Theor. Popul. Biol 53 175
[38]	Stone L, Hart D 1999 Theor. Popul. Biol 55 227
[39]	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]
[40]	Vandermeer J 1997 Ecol. Model. 95 311
[41]	Peng M S 2005 Chaos Solitons Fract. 24 1279
[42]	Peng MS 2005 Chaos Solitons Fract. 25 1123
[43]	Li T Y, Yorke J A 1975 Amer. Math. Monthly 82 985
[44]	Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[45]	Hu G, Xiao J H, Zheng Z G 2000 Chaos Control (Shanghai: Shanghai Scientific and Technological Education Publishing House) p8--20 (in Chinese) [胡岗, 萧井华, 郑志刚 2000 混沌控制(上海: 上海科技教育出版社) 第8---20页]

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Vol. 61, Issue 9 - 2012-05-05

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