Bifurcation analysis of macroscopic traffic flow model based on the influence of road conditions

Wenhuan Ai; Ting Zhang; Dawei Liu

Applications of Mathematics (2023)

  • Volume: 68, Issue: 4, page 499-534
  • ISSN: 0862-7940

Abstract

top
A macroscopic traffic flow model considering the effects of curves, ramps, and adverse weather is proposed, and nonlinear bifurcation theory is used to describe and predict nonlinear traffic phenomena on highways from the perspective of global stability of the traffic system. Firstly, the stability conditions of the model shock wave were investigated using the linear stability analysis method. Then, the long-wave mode at the coarse-grained scale is considered, and the model is analyzed using the reduced perturbation method to obtain the Korteweg-de Vries (KdV) equation of the model in the sub-stable region. In addition, the type of equilibrium points and their stability are discussed by using bifurcation analysis, and a theoretical derivation proves the existence of Hopf bifurcation and saddle-knot bifurcation in the model. Finally, the simulation density spatio-temporal and phase plane diagrams verify that the model can describe traffic phenomena such as traffic congestion and stop-and-go traffic in real traffic, providing a theoretical basis for the prevention of traffic congestion.

How to cite

top

Ai, Wenhuan, Zhang, Ting, and Liu, Dawei. "Bifurcation analysis of macroscopic traffic flow model based on the influence of road conditions." Applications of Mathematics 68.4 (2023): 499-534. <http://eudml.org/doc/299528>.

@article{Ai2023,
abstract = {A macroscopic traffic flow model considering the effects of curves, ramps, and adverse weather is proposed, and nonlinear bifurcation theory is used to describe and predict nonlinear traffic phenomena on highways from the perspective of global stability of the traffic system. Firstly, the stability conditions of the model shock wave were investigated using the linear stability analysis method. Then, the long-wave mode at the coarse-grained scale is considered, and the model is analyzed using the reduced perturbation method to obtain the Korteweg-de Vries (KdV) equation of the model in the sub-stable region. In addition, the type of equilibrium points and their stability are discussed by using bifurcation analysis, and a theoretical derivation proves the existence of Hopf bifurcation and saddle-knot bifurcation in the model. Finally, the simulation density spatio-temporal and phase plane diagrams verify that the model can describe traffic phenomena such as traffic congestion and stop-and-go traffic in real traffic, providing a theoretical basis for the prevention of traffic congestion.},
author = {Ai, Wenhuan, Zhang, Ting, Liu, Dawei},
journal = {Applications of Mathematics},
keywords = {macro traffic flow; curves; ramps; bifurcation analysis},
language = {eng},
number = {4},
pages = {499-534},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bifurcation analysis of macroscopic traffic flow model based on the influence of road conditions},
url = {http://eudml.org/doc/299528},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Ai, Wenhuan
AU - Zhang, Ting
AU - Liu, Dawei
TI - Bifurcation analysis of macroscopic traffic flow model based on the influence of road conditions
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 499
EP - 534
AB - A macroscopic traffic flow model considering the effects of curves, ramps, and adverse weather is proposed, and nonlinear bifurcation theory is used to describe and predict nonlinear traffic phenomena on highways from the perspective of global stability of the traffic system. Firstly, the stability conditions of the model shock wave were investigated using the linear stability analysis method. Then, the long-wave mode at the coarse-grained scale is considered, and the model is analyzed using the reduced perturbation method to obtain the Korteweg-de Vries (KdV) equation of the model in the sub-stable region. In addition, the type of equilibrium points and their stability are discussed by using bifurcation analysis, and a theoretical derivation proves the existence of Hopf bifurcation and saddle-knot bifurcation in the model. Finally, the simulation density spatio-temporal and phase plane diagrams verify that the model can describe traffic phenomena such as traffic congestion and stop-and-go traffic in real traffic, providing a theoretical basis for the prevention of traffic congestion.
LA - eng
KW - macro traffic flow; curves; ramps; bifurcation analysis
UR - http://eudml.org/doc/299528
ER -

References

top
  1. Ai, W.-H., Shi, Z.-K., Liu, D.-W., 10.1016/j.physa.2015.06.004, Physica A 437 (2015), 418-429. (2015) Zbl1400.90088MR3371710DOI10.1016/j.physa.2015.06.004
  2. Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y., 10.1103/PhysRevE.51.1035, Phys. Rev. E (3) 51 (1995), 1035-1042. (1995) DOI10.1103/PhysRevE.51.1035
  3. Cao, J. F., Han, C. Z., Fang, Y. W., Nonlinear Systems Theory and Application, Xi’an Jiao Tong University Press, Xi’an (2006), ISBN 7-5605-2140-1Chinese. (2006) 
  4. Carrillo, F. A., Delgado, J., Saavedra, P., Velasco, R. M., Verduzco, F., 10.1142/S0218127413501915, Int. J. Bifurcation Chaos Appl. Sci. Eng. 23 (2013), Article ID 1350191, 15 pages. (2013) Zbl1284.90012MR3158306DOI10.1142/S0218127413501915
  5. Chen, B., Sun, D., Zhou, J., Wong, W., Ding, Z., 10.1016/j.ins.2020.02.009, Inform. Sci. 529 (2020), 59-72. (2020) MR4093031DOI10.1016/j.ins.2020.02.009
  6. Cui, N., Chen, B., Zhang, K., Zhang, Y., Liu, X., Zhou, J., 10.1016/j.physa.2018.08.009, Physica A 513 (2019), 32-44. (2019) DOI10.1016/j.physa.2018.08.009
  7. Daganzo, C. F., Laval, J. A., 10.1016/j.trb.2004.10.004, Transp. Res., Part B 39 (2005), 855-863. (2005) DOI10.1016/j.trb.2004.10.004
  8. Delgado, J., Saavedra, P., 10.1142/S0218127415500649, Int. J. Bifurcation Chaos Appl. Sci. Eng. 25 (2015), Article ID 1550064, 18 pages. (2015) Zbl1317.34074MR3349898DOI10.1142/S0218127415500649
  9. Gupta, A. K., Dhiman, I., 10.1007/s11071-014-1693-6, Nonlinear Dyn. 79 (2015), 663-671. (2015) MR3302725DOI10.1007/s11071-014-1693-6
  10. Gupta, A. K., Katiyar, V. K., 10.1088/0305-4470/38/19/002, J. Phys. A, Math. Gen. 38 (2005), 4069-4083. (2005) Zbl1086.90013MR2145802DOI10.1088/0305-4470/38/19/002
  11. Gupta, A. K., Katiyar, V. K., 10.1016/j.physa.2005.12.036, Physica A 368 (2006), 551-559. (2006) DOI10.1016/j.physa.2005.12.036
  12. Gupta, A. K., Katiyar, V. K., 10.1016/j.physa.2006.03.061, Physica A 371 (2006), 674-682. (2006) DOI10.1016/j.physa.2006.03.061
  13. Gupta, A. K., Redhu, P., 10.1016/j.physleta.2013.06.009, Phys. Lett., A 377 (2013), 2027-2033. (2013) Zbl1297.90017MR3083138DOI10.1016/j.physleta.2013.06.009
  14. Gupta, A. K., Sharma, S., 10.1088/1674-1056/19/11/110503, Chin. Phys. B 19 (2010), Article ID 110503, 9 pages. (2010) DOI10.1088/1674-1056/19/11/110503
  15. Gupta, A. K., Sharma, S., 10.1088/1674-1056/21/1/015201, Chin. Phys. B 21 (2012), Article ID 015201, 15 pages. (2012) DOI10.1088/1674-1056/21/1/015201
  16. Igarashi, Y., Itoh, K., Nakanishi, K., Ogura, K., Yokokawa, K., 10.1103/PhysRevLett.83.718, Phys. Rev. Lett. 83 (1999), 718-721. (1999) DOI10.1103/PhysRevLett.83.718
  17. Igarashi, Y., Itoh, K., Nakanishi, K., Ogura, K., Yokokawa, K., 10.1103/PhysRevE.64.047102, Phys. Rev. E (3) 64 (2001), Article ID 047102. (2001) DOI10.1103/PhysRevE.64.047102
  18. Jiang, R., Wu, Q., Zhu, Z., 10.1103/PhysRevE.64.017101, Phys. Rev. E (3) 64 (2001), Article ID 017101. (2001) MR2998582DOI10.1103/PhysRevE.64.017101
  19. Jiang, R., Wu, Q.-S., Zhu, Z.-J., 10.1016/S0191-2615(01)00010-8, Transp. Res., Part B 36 (2002), 405-419. (2002) DOI10.1016/S0191-2615(01)00010-8
  20. Kerner, B. S., Konhäuser, P., 10.1103/PhysRevE.48.R2335, Phys. Rev. E (3) 48 (1993), 2335-2338. (1993) DOI10.1103/PhysRevE.48.R2335
  21. Kuznetsov, Y. A., 10.1007/978-0-387-22710-8_5, Elements of Applied Bifurcation Theory Applied Mathematical Sciences 112. Springer, New York (1998), 151-194. (1998) Zbl0914.58025MR1711790DOI10.1007/978-0-387-22710-8_5
  22. Lei, L., Wang, Z., Wu, Y., 10.32604/cmes.2022.019855, CMES, Comput. Model. Eng. Sci. 131 (2022), 1815-1830. (2022) DOI10.32604/cmes.2022.019855
  23. Ling, D., Jian, X. P., Stability and bifurcation characteristics of a class of nonlinear vehicle following model, J. Traffic and Transportation Engineering and Information 7 (2009), 6-11. (2009) 
  24. Ma, G., Ma, M., Liang, S., Wang, Y., Guo, H., 10.1016/j.physa.2020.125303, Physica A 562 (2021), Article ID 125303, 12 pages. (2021) Zbl07542618MR4157710DOI10.1016/j.physa.2020.125303
  25. Ma, G., Ma, M., Liang, S., Wang, Y., Zhang, Y., 10.1016/j.cnsns.2020.105221, Commun. Nonlinear Sci. Numer. Simul. 85 (2020), Article ID 105221, 10 pages. (2020) Zbl1452.65169MR4065383DOI10.1016/j.cnsns.2020.105221
  26. Meng, X. P., Yan, L. Y., 10.1002/asjc.1505, Asian J. Control 19 (2017), 1844-1853. (2017) Zbl1386.93217MR3704494DOI10.1002/asjc.1505
  27. Orosz, G., Wilson, R. E., Krauskopf, B., 10.1103/PhysRevE.70.026207, Phys. Rev. E (3) 70 (2004), Article ID 026207, 10 pages. (2004) MR2129214DOI10.1103/PhysRevE.70.026207
  28. Redhu, P., Gupta, A. K., 10.1016/j.cnsns.2015.03.015, Commun. Nonlinear Sci. Numer. Simul. 27 (2015), 263-270. (2015) Zbl1457.93068MR3341560DOI10.1016/j.cnsns.2015.03.015
  29. Zeng, J., Qian, Y., Xu, D., Jia, Z., Huang, Z., 10.1155/2014/218465, Math. Probl. Eng. 2014 (2014), Article ID 218465, 6 pages. (2014) Zbl1407.90103MR3166824DOI10.1155/2014/218465
  30. Zhai, C., Wu, W., 10.1007/s11071-018-4318-7, Nonlinear Dyn. 93 (2018), 2185-2199. (2018) DOI10.1007/s11071-018-4318-7
  31. Zhai, C., Wu, W., 10.1142/S0129183119500736, Int. J. Mod. Phys. C 30 (2019), Article ID 1950073, 14 pages. (2019) MR4015821DOI10.1142/S0129183119500736
  32. Zhai, C., Wu, W., 10.1142/S0217984919502737, Mod. Phys. Lett. B 33 (2019), Article ID 1950273, 16 pages. (2019) MR3993691DOI10.1142/S0217984919502737
  33. Zhai, C., Wu, W., 10.1142/S0129183120500898, Int. J. Mod. Phys. C 31 (2020), Article ID 2050089, 16 pages. (2020) MR4119105DOI10.1142/S0129183120500898
  34. Zhai, C., Wu, W., 10.1142/S0217984920500712, Mod. Phys. Lett. B 34 (2020), Article ID 2050071, 15 pages. (2020) MR4068029DOI10.1142/S0217984920500712
  35. Zhai, C., Wu, W., 10.1142/S0217984921500548, Mod. Phys. Lett. B 35 (2021), Article ID 2150054, 15 pages. (2021) MR4202802DOI10.1142/S0217984921500548
  36. Zhai, C., Wu, W., 10.1016/j.cnsns.2020.105667, Commun. Nonlinear Sci. Numer. Simul. 95 (2021), Article ID 105667, 18 pages. (2021) Zbl1456.82635MR4192012DOI10.1016/j.cnsns.2020.105667
  37. Zhang, P., Xue, Y., Zhang, Y.-C., Wang, X., Cen, B.-L., 10.1142/S0217984920502176, Mod. Phys. Lett. B 34 (2020), Article ID 2050217, 18 pages. (2020) MR4128734DOI10.1142/S0217984920502176

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.