Dynamical transition in a coupled-map lattice model of a recurrent bus

We investigate the dynamical behavior of a recurrent bus on a cyclic route with many bus stops. The maximum capacity of the bus is taken into account to study the dynamics of a waiting queue at a bus stop. We build a coupled-map lattice model of a cyclic bus. The recurrence time (one period) on each bus stop depends on the number M of bus stops, the maximum capacity Wmax of the bus, and the stopping time for new passengers to board the bus. The coupled-map lattice of all bus stops is approximated with a single nonlinear map of the average of all the stops. It is shown that the dynamical phase transition between the queuing and schedule-time phases occurs with varying the initial period. When the initial period is higher than a critical value, some persons waiting at a bus stop may not be able to board the bus because the bus is full. The waiting line at each bus stop increases with time, forming a queue. In contrast, if the initial period is less than the critical value, all waiting people at a bus stop can board the bus since the bus is not full. By selecting the combination of the number of bus stops, the maximum capacity of the bus and the initial period, one can operate the bus on a scheduled time without queues for an infinite time.