Based on the new macroscopic two-velocity difference model, this paper analyzes the linear stability of the new model and studies the nonlinear bifurcation theory. First, the linear stability analysis method is used to study the stability conditions of the shock wave in the model. Then, considering the long wave model in the coarse-grained scale, the reduced perturbation method is used to analyze the characteristics of the traffic flow in the metastable region, and the solitary wave solution of the Korteweg-de Vries (KdV) equation in the metastable region is derived. In addition, by using the bifurcation analysis method, the type, and stability of the equilibrium solution are discussed and the existing conditions of the saddle-node bifurcation are proven. Then, taking the saddle-node bifurcation as the starting point, we draw the density space-time diagram and phase plane diagram of the system. It is proven that the newly proposed model can describe complex traffic phenomena such as stop-and-go and sudden changes in stability, which is of great help to solve traffic congestion.

本文基于新的宏观二速差模型，分析了新模型的线性稳定性，研究了非线性分岔理论。首先，采用线性稳定性分析方法研究模型中冲击波的稳定性条件。然后，在粗粒度尺度下考虑长波模型，采用缩减摄动法分析亚稳区交通流的特征，得到Korteweg-de Vries (KdV)方程的孤波解在亚稳区是派生的。此外，利用分岔分析方法，讨论了平衡解的类型和稳定性，证明了鞍结分岔存在的条件。然后，以鞍结点分叉为起点，我们绘制了系统的密度时空图和相平面图。事实证明，新提出的模型可以描述走走停停、稳定性突变等复杂交通现象，对解决交通拥堵有很大帮助。