Dynamical transitions to chaotic and periodic motions of two shuttle buses

We study the dynamical behavior of two shuttle buses on a route where two buses shuttle between a starting point and a destination. We present the dynamical model for the two shuttle buses. When a bus catches another, the bus restarts after delay time Tmin. The distinct dynamical states (the regular, periodic, and chaotic motions) are found by varying both loading parameter μγ and delay time Tmin. It is shown that dynamical transitions occur from the regular motion, through multiply periodic motions, to the chaotic motion, with increase of loading parameter μγ and delay time Tmin. Liapunov exponent and the time-series plot are calculated for the chaotic motion.