Mathematical Modelling of Oscillatory Dynamics in Circular Traffic Systems

TL;DR

This paper develops an analytical mathematical model for oscillatory traffic behavior on circular tracks, highlighting how driver responses and delays cause standing waves, with implications for autonomous vehicle control.

Contribution

It introduces a novel delay differential equation framework incorporating stochastic and cognitive factors, providing a purely analytical stability analysis without simulations.

Findings

Identifies critical thresholds for harmonic oscillations

Delineates safe following distance bounds

Reveals hysteresis effects in driver correction behavior

Abstract

This paper presents a rigorous analytical model of traffic dynamics on a circular track, demonstrating the emergence of standing oscillations resulting from microscopic driver behaviour, delay responses, and proximity pressure. Without relying on simulation, we derive a series of coupled delay differential equations to model vehicular interactions. By introducing a mnemonic-based symbolic system, we establish a mathematical framework incorporating stochastic initial conditions, non-uniform reaction times, and cognitive lag. A full linear stability analysis is conducted using Fourier decomposition and modal perturbation techniques. Our results identify critical thresholds for harmonic induction, delineate the bounds of safe following distances, and reveal hysteresis in driver overcorrection. The analysis concludes with implications for autonomous vehicle control and potential suppression strategies for oscillatory instability. All derivations are purely symbolic and analytically proven.