Chaotic and periodic motions of a cyclic bus induced by speedup

We study the effect of speedup on the dynamical behavior of a single cyclic bus in a bus system with many bus stops. We present a nonlinear-map model of a cyclic bus to take into account the speedup. When the cyclic bus is delayed, the bus speeds up to retrieve the delay. It is found that the cyclic bus exhibits chaotic motion with increasing speedup. The chaotic motion depends on both speedup and the number of bus stops. Also, it is shown that the dynamical transition between the chaotic and periodic motions occurs with the increase of bus stops. The dependence of the recurrence time (one period) and Liapunov exponent on both speedup and the number of bus stops is calculated for distinct dynamic states. It is shown that the speedup has a significant effect on the bus motion. For a piecewise linear-map model, the cyclic bus does not exhibit the chaotic motion but a complex oscillatory motion with multiple periods occurs.