Transitions to chaos of a shuttle bus induced by continuous speedup

We study the effect of continuous speedup on the dynamical behavior of a single shuttle bus. We present a nonlinear map model of a shuttle bus to take into account the continuous speedup. When the shuttle bus is delayed, the bus speeds up continuously to overcome the delay. If the shuttle bus speeds up discontinuously, the dynamics is described in terms of the piecewise linear map and the bus then exhibits the complex oscillatory motion with multiple periods. For the continuous speedup, it is found that the bus exhibits chaotic motion by the pitchfork bifurcation. The chaotic motion depends on loading parameter, maximum speed, and speedup rate. The dependence of the recurrence time (one period) and Liapunov exponent on these parameters is calculated. It is shown that the speedup rate has the significant effect on the bus motion.