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We present the model for the vehicular traffic in a city traffic network controlled by traffic lights. The vehicular motion on a selected path is described in terms of the nonlinear map. We study the dynamic behavior of vehicular traffic through the series of traffic lights on selected paths in two-dimensional (2d) traffic network. The vehicle moves through the series of signals on a path selected by the driver. The selected path is one of the straight, zigzag, and random paths in city traffic network. The vehicular traffic exhibits the very complex behavior with varying selected path, cycle time, vehicular speed, and vehicular density. The dependence of the arrival time on the parameters is clarified for city traffic.