Synchronization in bus systems with partially overlapping routes

In an increasingly interconnected world, understanding congestion-related phenomena in transportation and their underlying mechanisms is crucial for improving efficiency. As the transportation system becomes denser, different modes of transportation have more opportunities to interact with each other, giving rise to emergent dynamics that simple models cannot explain. In this study, we investigate the synchronized motion of indirectly coupled transportation modes. We develop a numerical simulation model on a one-dimensional periodic lattice, where each point represents a bus station. In this system, two types of buses operate: multiple local buses with nonoverlapping routes, each serving a specific zone, and a single global bus that partially overlaps with the routes of the local buses. We perform numerical simulations to examine how close the arrival times of these buses are to each other-that is, how synchronized their motions are. When the number of zones is two, three, or five, robust synchronization occurs not only between the global bus and the local buses but also among the local buses themselves. In contrast, no synchronization is found for other numbers of zones. We developed a mathematical model using self-consistent equations and found that two distinct arrival patterns at the terminals must be considered. A stability analysis reveals which pattern is ultimately realized in the simulations. Our results show that transportation modes can exhibit coherent motion even when sharing only partial or no direct route overlaps. This outcome highlights that emergent behavior depends not only on local interactions but is also strongly shaped by the system's overall structural configuration.