Synchronization in Traffic Dynamics: Mechanisms of Hysteresis

TL;DR

This paper explores the dynamical mechanisms of hysteresis in traffic flow using a second-order differential model, revealing how time-delay and collision metrics relate to synchronization and energy dissipation.

Contribution

It introduces a theoretical framework linking hysteresis, time-delay, and collision metrics to traffic synchronization phenomena, with phase diagrams bridging different spatial representations.

Findings

Identification of hysteresis patterns in phase space trajectories.

Demonstration that time-delay and time-to-collision are metrics of zero dissipation.

Development of a phase diagram connecting traffic behaviors across different spaces.

Abstract

Starting from a second-order linear differential equation, we analyze the dynamical mechanisms of no behavior pattern (pure response), reaction and anticipation behaviors in traffic. As an emergence of the underlying dynamical evolution, the periodic evolution trajectories (3D hysteresis) in phase space ($v_i, v_j, d_{ji}$) exhibit fascinating characters. We investigate the emerging Time-Delay ($TD$) phenomena and the resulting analytical hysteresis, an equal frequency sets of Lissajous figures. By quantifying energy dissipation through individual and system perspectives, we demonstrate that $TD$ and Time-To-Collision ($TTC$) are direct metrics of zero-dissipation under equilibrium and synchronization states. Finally, a phase diagram based on $TD$ and $TTC$ is developed to bridge the dynamical behaviors in traffic across $\mathbb{R}^1$ and $\mathbb{R}^2$ spaces. Our results provide a theoretical foundation upon which many obscure mechanisms become self-evident, such as the $TD$-induced flip of the hysteresis (clockwise to counterclockwise in FD) and the crossed hysteresis, etc.