Synchronization in Traffic Dynamics: Mechanisms of Hysteresis

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.