We study the dynamical behavior of vehicular traffic through a series of traffic signals. The vehicular traffic is controlled with the use of the cycle time generated by a logistic map. Each signal changes periodically with a cycle time, and the cycle time varies from signal to signal. The nonlinear dynamic model of the vehicular motion is presented by a nonlinear map including the logistic map. The vehicular traffic exhibits very complex behavior on varying both the cycle time and the logistic-map parameter a. For a>3, the arrival time shows a linear dependence on the cycle time. Also, the dependence of vehicular motion on parameter a is clarified.
Physics, other sciences and technologies meet at the frontier area of interdisciplinary research. The concepts and techniques of physics are being applied to such complex systems as transportation systems. Recently, transportation problems have attracted much attention in the fields of physics [1], [2], [3], [4] and [5]. Traffic flow, pedestrian flow, and bus-route problems have been studied from a point of view of statistical mechanics and nonlinear dynamics [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] and [19]. The jams, chaos, and pattern formation are typical signatures of the complex behavior of transportation.
Mobility is nowadays one of the most significant ingredients of a modern society. In urban traffic, vehicles are controlled by traffic signals to give priority to one road where roads meet at crossings. In real traffic, the vehicular traffic flow depends highly on the control of traffic signals. Optimizing traffic lights for city traffic has been studied by using the CA traffic model and the optimal velocity model [20] and [21]. The effect of signal control strategy on vehicular traffic has been clarified. It has been shown that city traffic controlled by traffic signals can be reduced to a simpler problem of a single-lane highway with a few signals. There have been studies of vehicular traffic controlled by a few traffic signals. It has been concluded that the periodic traffic does not depend on the number of traffic lights.
Very recently, a few studies have been made on the traffic flow of vehicles moving through an infinite series of traffic lights with the same interval. The effect of cycle time on vehicular traffic has been clarified [22], [23], [24], [25], [26], [27] and [28].
Generally, traffic lights are controlled by either synchronized or green-wave strategies. In the synchronized strategy, all the signals change simultaneously and periodically; the phase shift has the same value for all signals. In the green-wave strategy, the signal changes with a certain time delay between the signal phases of two successive intersections. The change of traffic lights propagates backward like a green wave. Thus, the vehicular traffic flow is controlled by varying the phase shift of the signals. More generally, the traffic signal can be controlled by means of the phase shift (offset time), cycle time, and split time.
The operator will be able to control the traffic signal by the use of another strategy. Specifically, one can manage the cycle times of signals. One will be able to control the vehicular traffic by changing the cycle time from signal to signal. Until now, vehicular traffic flow has been studied only in such a case that all signals have the same value for the cycle time. The study of vehicular traffic through a sequence of signals with the cycle time varying from signal to signal has been unknown.
In this paper, we study vehicular traffic flow through an infinite series of signals with cycle time varying from signal to signal. We apply a logistic map to the signal’s strategy. We control the cycle time by using the logistic map. We present a nonlinear dynamic model for the vehicular motion through the series of traffic signals. We investigate the dynamical behavior of the vehicular traffic. We clarify the dependence of the vehicular motion through the sequence of signals on both the cycle time and the logistic-map parameter.
We have studied vehicular motion through a sequence of traffic signals with the cycle time generated by a logistic map where the cycle time varies from signal to signal. We have presented a nonlinear map model for the nonlinear dynamics of the vehicular traffic. We have derived the relationships between the arrival time, the standard cycle time, and the logistic-map parameter. We have clarified the dependence of arrival time on both the standard cycle time and the logistic-map parameter. We have shown that the operator is able to control the vehicular traffic successfully by the logistic map via the cycle time.
This study of the control of the cycle time by a logistic map is the first, and this method will be useful for controlling vehicular traffic with traffic signals.
