Phase Plane Analysis Method of Nonlinear Traffic Phenomena

Wenhuan Ai,Dawei Liu,Zhongke Shi

doi:10.1155/2015/603536

Wenhuan Ai, Dawei Liu + Show 1 more

Open Access

https://doi.org/10.1155/2015/603536

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Abstract

A new phase plane analysis method for analyzing the complex nonlinear traffic phenomena is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the analysis in phase plane. According to the new model, various traffic phenomena, such as the well-known shock waves, rarefaction waves, and stop-and-go waves, are analyzed in the phase plane. From the phase plane diagrams, we can see the relationship between traffic jams and system instability. So the problem of traffic flow could be converted into that of system stability. The results show that the traffic phenomena described by the new method is consistent with that described by traditional methods. Moreover, the phase plane analysis highlights the unstable traffic phenomena we are chiefly concerned about and describes the variation of density or velocity with time or sections more clearly.

Highlights

It is really necessary to study various traffic phenomena for alleviating and preventing traffic jams
When the shock wave and rarefaction wave appears, the curves in phase plane diagrams change from one equilibrium point to another
The corresponding phase plane diagram of (σ, ∂σ/∂x) in Figure 5(c) obviously shows that the inner circle of the current time turns to the outer ring of the second and the traffic become congested with time

Summary

Introduction

It is really necessary to study various traffic phenomena for alleviating and preventing traffic jams. Some researchers have described shock wave and rarefaction wave with the use of new models they proposed [1,2,3]. They treat each individual vehicle as a particle and regard traffic as a complex system of interacting particles Based on these models, researchers can use a lot of system simulation methods to analyze the traffic phenomena. Researchers can use a lot of system simulation methods to analyze the traffic phenomena These methods cannot completely describe the complex traffic flow phenomena in phase plane from a system stability perspective.

Variable Substitution Based on Jiang’s Model

Analysis of the Stop-and-Go Traffic Phenomenon Using the New Method

Analysis of Shock and Rarefaction Waves Using the New Method

Conclusions