Synchronization in Traffic Dynamics: Mechanisms of Hysteresis

TL;DR

The paper develops a unified analytical framework for hysteresis in traffic dynamics by examining four second-order models built on a constant time-headway strategy. It shows that the resulting 3D oscillations form ellipses in phase space, with 2D hysteresis curves emerging as projections of these trajectories; the area of the loops quantifies kinetic-energy dissipation and motivates two concise metrics, Time-Delay $($TD$)$ and Time-To-Collision $($TTC$)$, to characterize synchronization. By deriving transfer functions, stability criteria, and phase relationships, the work reveals how reaction and anticipation differently shape hysteresis and energy dissipation, and it presents a TD–TTC phase diagram that maps to real-world traffic behavior across 1D and 2D settings. The findings offer a theoretical foundation for understanding and predicting complex traffic phenomena such as hysteresis flips and crossed hysteresis, with implications for traffic control and safety analysis.

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.

Figures (11)

• Figure 1: Bode diagram of transfer function ($G(j\omega)$) in the four models.
• Figure 2: Phase diagrams for string stability in CosForce and FVD models.
• Figure 3: Hysteresis characteristics in the four models. Parameter settings for the input wave function: $A = 0.8, \omega = 1$.
• Figure 4: 3D hysteresis of the four models.
• Figure 5: Dynamic evolution of $TTC$ in phase trajectory.