Delay transition of a recurrent bus on a circular route

We study the behavior of a recurrent bus on a circular route with many bus stops. We present the basic model determining the recurrent time (one period) on each bus stop. The recurrent times on M bus stops are given by M simultaneous nonlinear equations. It is found that the dynamical phase transition between the delay and schedule-time phases occurs with varying the initial period. The dynamical transition depends on number M of bus stops. When M is larger than the critical value, the recurrent time diverges with the number of rotation. The recurrent time is described, approximately, in terms of a nonlinear map. The schedule-time phase is determined by the basin of attraction.