Stability of anti-bunched buses and local unidirectional Kuramoto oscillators

TL;DR

This paper models bus bunching and unidirectional Kuramoto oscillators to understand how stable anti-bunched states can be achieved, revealing that such states naturally occur in systems with at least five oscillators.

Contribution

It introduces a novel analogy between bus bunching phenomena and Kuramoto oscillators, demonstrating conditions for stable anti-bunched states without external forcing.

Findings

Stable anti-bunched states exist in Kuramoto oscillators with five or more units.

A no-boarding policy acts as a stabilizing force in bus systems.

The bus system and oscillator model share similar synchronization dynamics.

Abstract

Inspired by our recent work that relates bus bunching as a phenomenon of synchronisation of phase oscillators, we construct a model of Kuramoto oscillators that follows an analogous interaction mechanism of local unidirectional coupling. In the bus loop system, we can introduce a no-boarding policy as a form of kicking force to achieve a stable staggered (anti-bunched) state. For Kuramoto oscillators, it turns out that such stable anti-bunched states can exist (without any additional kicking force) if the number of oscillators are at least five. This correspondence between the bus loop system and the local unidirectional Kuramoto oscillators leads to the insight on how the bus loop system can remain staggered.