Analytical Calculation of Critical Perturbation Amplitudes and Critical Densities by Non-Linear Stability Analysis of a Simple Traffic Flow Model

TL;DR

This paper presents an analytical approach to determine critical perturbation amplitudes and densities in a traffic flow model, elucidating the conditions leading to traffic breakdowns due to non-linear instabilities.

Contribution

It introduces an analytical method for studying non-linear stability in traffic models, complementing previous numerical approaches.

Findings

Analytical results match numerical simulations well.

Identified critical perturbation amplitudes causing traffic breakdown.

Demonstrated metastability regions in traffic density regimes.

Abstract

Driven many-particle systems with nonlinear interactions are known to often display multi-stability, i.e. depending on the respective initial condition, there may be different outcomes. Here, we study this phenomenon for traffic models, some of which show stable and linearly unstable density regimes, but areas of metastability in between. In these areas, perturbations larger than a certain critical amplitude will cause a lasting breakdown of traffic, while smaller ones will fade away. While there are common methods to study linear instability, non-linear instability had to be studied numerically in the past. Here, we present an analytical study for the optimal velocity model with a stepwise specification of the optimal velocity function and a simple kind of perturbation. Despite various approximations, the analytical results are shown to reproduce numerical results very well.