Dynamical behavior of N shuttle buses not passing each other: chaotic and periodic motions

We study the dynamical behavior of shuttle buses on a route where N buses (N⩾2) shuttle between a starting point and a destination without passing. We present the dynamical model of N shuttle buses to take into account the interactions between buses and passengers. The model is described in terms of loading parameter, delay (restart) time, and bus's number. It is found that the distinct dynamical states (the regular, periodic, and chaotic motions) appear by varying both loading parameter and delay time. Also, the dynamical states depend highly on the number of buses. The dynamical transitions among the distinct dynamical states and the phase diagram are shown. The recurrence time (one period) and Liapunov exponent are calculated for distinct dynamic states. The region of periodic motion scales as ΔTmin∝N−1.72 where N is the number of buses.