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宏观交通流模型的余维2分岔分析

CODIMENSION-2 BIFURCATION ANALYSIS OF MACROSCOPIC TRAFFIC FLOW MODEL

  • 摘要: 交通流特性是混合交通流建模的一个重要因素. 交通流模型中的分岔现象是导致复杂交通现象的因素之一. 交通流的分岔, 涉及复杂的动力学特征且研究较少. 因此, 提出了一个最优速度模型来研究驾驶员记忆对驾驶行为的影响. 基于带有记忆的最优速度连续交通流模型, 利用非线性动力学, 分析和预测了复杂交通现象. 推导了鞍结 (LP) 分岔存在条件, 并通过数值计算得到了余维1 Hopf (H) 分岔、LP分岔和同宿轨 (HC) 分岔以及余维2广义Hopf (GH) 分岔、尖点 (CP) 分岔和Bogdanov-Takens (BT) 分岔等多种分岔结构. 根据双参数分岔区域的特点, 研究了记忆参数对单参数分岔结构的影响, 分析了不同分岔结构对交通流的影响, 并用相平面描述了平衡点附近轨迹的变化特征. 选择Hopf分岔和鞍结分岔作为密度演化的起点, 描述了均匀流、稳定和不稳定的拥挤流以及走走停停现象. 结果表明, 驾驶员记忆对交通流的稳定性有重要影响; 动力学行为很好地解释了交通拥堵现象; 考虑余维2分岔的影响, 能更好地理解交通拥堵产生的根源, 并为制定有效抑制拥堵的方法提供一定的理论依据.

     

    Abstract: The traffic flow characteristics are an important factor in mixed traffic flow modeling. The bifurcation in the traffic flow model is one of the issues related to the complex traffic phenomena. The bifurcation phenomenon of traffic flow models involves complex dynamic characteristics and is rarely studied. Therefore, an optimal velocity model is proposed to study the effects of driver’s memory on driving behavior. Based on the optimal velocity continuous traffic flow model with memory, we analyze and predict complex traffic phenomena by using nonlinear dynamics. The conditions for the existence of LP bifurcation are derived. We numerically obtain codim-1 Hopf (H) bifurcation, LP bifurcation and homoclinic (HC) bifurcation, and codim-2 generalized Hopf (GH) bifurcation, cusp (CP) bifurcation and Bogdanov-Takens (BT) bifurcation. According to the characteristics of two-parameter bifurcation regions, the influence of memory parameters on the one-parameter bifurcation structures is studied, and the influence of different bifurcation structures on traffic flow is analyzed. The phase plane is used to describe the variational characteristics of the trajectories near the equilibrium point. Selecting the Hopf bifurcation and saddle-node bifurcation as the starting point of density evolution, we describe the uniform flow, stable and unstable crowded flow and stop-and-go phenomena. Further, these outcomes can improve the understanding of go-and-stop waves and local clusters observed on highways. The results show that the driver’s memory plays an important role in the stability of the traffic flow. Dynamic behavior can well explain the complex phenomenon of congested traffic. The source of traffic congestion can be better understood by considering the impact of codim-2 bifurcation. The results in this paper can provide some theoretical methods for the suppress traffic congestion.

     

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