经典跟车模型的稳定性、收敛性及霍普夫分岔分析
Stability, convergence and Hopf bifurcation analyses of the classical car-following model
AI总结：
本文研究了延迟反馈对经典跟车模型动态特性的影响，分析了无延迟、小延迟和任意延迟情况下的稳定性条件，揭示了交通流从局部稳定到不稳定过渡通过霍普夫分岔产生极限环的现象，并探讨了非振荡收敛条件、收敛速率以及霍普夫分岔类型和极限环渐近轨道稳定性。
摘要：
反应延迟在确定车辆编队在直路上行驶的定性动态特性中起着重要作用。在本文中，我们研究了延迟反馈对经典跟车模型（CCFM）动态特性的影响。具体来说，我们分析了无延迟、小延迟和任意延迟情况下的CCFM。首先，我们推导出无延迟和小延迟情况下CCFM局部稳定的充分条件。接着，我们推导出任意延迟情况下CCFM局部稳定的必要且充分条件。然后我们证明交通流从局部稳定到不稳定状态的过渡通过霍普夫分岔发生，从而导致系统动态中的极限环。从物理角度来看，这些极限环表现为高速公路上的回波传播拥堵波。在人类驾驶车辆的背景下，我们的工作提供了关于反应延迟对交通拥堵产生和演变影响的表征性见解。在自动驾驶车辆的背景下，我们的工作有可能为自动驾驶汽车中运行的控制算法提供设计指南，以避免不利现象。具体来说，设计避免突突行驶的控制算法至关重要。因此，我们推导出CCFM非振荡收敛的必要且充分条件。接下来，我们刻画了CCFM的收敛速率，并探讨了局部稳定性、非振荡收敛性和CCFM收敛速率之间的相互作用。进一步，为了更好地理解系统动态中的振荡，我们利用庞加莱正常形式和中心流形理论来刻画霍普夫分岔的类型以及极限环的渐近轨道稳定性。分析还辅以稳定性图、分岔图和MATLAB仿真。
英文摘要：
Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles traversing a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Classical Car-Following Model (CCFM). Specifically, we analyze the CCFM in no delay, small delay and arbitrary delay regimes. First, we derive a sufficient condition for local stability of the CCFM in no-delay and small-delay regimes using. Next, we derive the necessary and sufficient condition for local stability of the CCFM for an arbitrary delay. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation, thus resulting in limit cycles in system dynamics. Physically, these limit cycles manifest as back-propagating congestion waves on highways. In the context of human-driven vehicles, our work provides phenomenological insight into the impact of reaction delays on the emergence and evolution of traffic congestion. In the context of self-driven vehicles, our work has the potential to provide design guidelines for control algorithms running in self-driven cars to avoid undesirable phenomena. Specifically, designing control algorithms that avoid jerky vehicular movements is essential. Hence, we derive the necessary and sufficient condition for non-oscillatory convergence of the CCFM. Next, we characterize the rate of convergence of the CCFM, and bring forth the interplay between local stability, non-oscillatory convergence and the rate of convergence of the CCFM. Further, to better understand the oscillations in the system dynamics, we characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the limit cycles using Poincare normal forms and the center manifold theory. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations.
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