Self-similar behavior of a single vehicle through periodic traffic lights

We study the dynamical behavior of a single vehicle moving through a sequence of traffic lights on a single-lane highway, where the traffic lights turn on and off periodically with the synchronized or green-wave strategies. The dynamical model of the vehicle controlled by traffic lights is expressed in terms of the nonlinear map. The tour time of the vehicle exhibits a complex behavior, interacting with the sequence of traffic lights. The dynamical behavior of signal traffic at a low density changes at ts=l/v+τ, where l is the distance between the traffic lights, v the mean speed of the vehicle, and τ the delay of traffic light at the green-wave strategy. When the cycle time ts of traffic lights is less than (l/v+τ), the vehicle exhibits the self-similar behavior, while it shows the periodic behavior for ts>(l/v+τ). The characteristics of the self-similar behavior is clarified.