Complex motions of shuttle buses by speed control

We study the dynamical behavior of shuttle buses on a route where N buses shuttle between an origin and a destination repeatedly. We present the dynamic model of shuttle buses to take into account the speed control depending on time headway between buses. The model is described by N-dimensional map governed by loading parameter and speed's variation rate. It is found that the distinct dynamical states (the regular, periodic, and chaotic motions) appear by varying both parameters. Also, the dynamical states depend on the number of buses. The dynamical transitions among the distinct dynamical states are investigated. The recurrence time (one period), Liapunov exponent, and return map are calculated for distinct dynamic states.