dorsal/arxiv
View SchemaSolvable Optimal Velocity Models and Asymptotic Trajectory
| Authors | K. Nakanishi, K. Itoh, Y. Igarashi, M. Bando |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9610009 |
| URL | https://arxiv.org/abs/patt-sol/9610009 |
| DOI | 10.1103/PhysRevE.55.6519 |
Abstract
In the Optimal Velocity Model proposed as a new version of Car Following Model, it has been found that a congested flow is generated spontaneously from a homogeneous flow for a certain range of the traffic density. A well-established congested flow obtained in a numerical simulation shows a remarkable repetitive property such that the velocity of a vehicle evolves exactly in the same way as that of its preceding one except a time delay $T$. This leads to a global pattern formation in time development of vehicles' motion, and gives rise to a closed trajectory on $\Delta x$-$v$ (headway-velocity) plane connecting congested and free flow points. To obtain the closed trajectory analytically, we propose a new approach to the pattern formation, which makes it possible to reduce the coupled car following equations to a single difference-differential equation (Rondo equation). To demonstrate our approach, we employ a class of linear models which are exactly solvable. We also introduce the concept of ``asymptotic trajectory'' to determine $T$ and $v_B$ (the backward velocity of the pattern), the global parameters associated with vehicles' collective motion in a congested flow, in terms of parameters such as the sensitivity $a$, which appeared in the original coupled equations.
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"abstract": "In the Optimal Velocity Model proposed as a new version of Car Following\nModel, it has been found that a congested flow is generated spontaneously from\na homogeneous flow for a certain range of the traffic density. A\nwell-established congested flow obtained in a numerical simulation shows a\nremarkable repetitive property such that the velocity of a vehicle evolves\nexactly in the same way as that of its preceding one except a time delay $T$.\nThis leads to a global pattern formation in time development of vehicles\u0027\nmotion, and gives rise to a closed trajectory on $\\Delta x$-$v$\n(headway-velocity) plane connecting congested and free flow points. To obtain\nthe closed trajectory analytically, we propose a new approach to the pattern\nformation, which makes it possible to reduce the coupled car following\nequations to a single difference-differential equation (Rondo equation). To\ndemonstrate our approach, we employ a class of linear models which are exactly\nsolvable. We also introduce the concept of ``asymptotic trajectory\u0027\u0027 to\ndetermine $T$ and $v_B$ (the backward velocity of the pattern), the global\nparameters associated with vehicles\u0027 collective motion in a congested flow, in\nterms of parameters such as the sensitivity $a$, which appeared in the original\ncoupled equations.",
"arxiv_id": "patt-sol/9610009",
"authors": [
"K. Nakanishi",
"K. Itoh",
"Y. Igarashi",
"M. Bando"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevE.55.6519",
"title": "Solvable Optimal Velocity Models and Asymptotic Trajectory",
"url": "https://arxiv.org/abs/patt-sol/9610009"
},
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